Properties

Label 40-384e20-1.1-c5e20-0-3
Degree $40$
Conductor $4.860\times 10^{51}$
Sign $1$
Analytic cond. $6.16361\times 10^{35}$
Root an. cond. $7.84776$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 22·9-s + 6.11e3·23-s − 2.36e4·25-s + 2.61e4·47-s + 1.27e5·49-s − 7.22e4·71-s − 6.12e4·73-s + 2.26e4·81-s + 9.28e4·97-s + 1.39e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 3.44e6·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 0.0905·9-s + 2.40·23-s − 7.56·25-s + 1.72·47-s + 7.58·49-s − 1.70·71-s − 1.34·73-s + 0.383·81-s + 1.00·97-s + 8.68·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 9.28·169-s + 2.54e−6·173-s + 2.33e−6·179-s + 2.26e−6·181-s + 1.98e−6·191-s + 1.93e−6·193-s + 1.83e−6·197-s + 1.79e−6·199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{140} \cdot 3^{20}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(40\)
Conductor: \(2^{140} \cdot 3^{20}\)
Sign: $1$
Analytic conductor: \(6.16361\times 10^{35}\)
Root analytic conductor: \(7.84776\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((40,\ 2^{140} \cdot 3^{20} ,\ ( \ : [5/2]^{20} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(18.18067038\)
\(L(\frac12)\) \(\approx\) \(18.18067038\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 22 T^{2} - 7385 p T^{4} - 81272 p^{2} T^{6} - 1029758 p^{6} T^{8} + 4685668 p^{10} T^{10} - 1029758 p^{16} T^{12} - 81272 p^{22} T^{14} - 7385 p^{31} T^{16} + 22 p^{40} T^{18} + p^{50} T^{20} \)
good5 \( ( 1 + 11818 T^{2} + 71488773 T^{4} + 301719010488 T^{6} + 1020428065064178 T^{8} + 3187442823228138492 T^{10} + 1020428065064178 p^{10} T^{12} + 301719010488 p^{20} T^{14} + 71488773 p^{30} T^{16} + 11818 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
7 \( ( 1 - 63774 T^{2} + 2508631325 T^{4} - 71829091413864 T^{6} + 1643928188403720658 T^{8} - \)\(30\!\cdots\!84\)\( T^{10} + 1643928188403720658 p^{10} T^{12} - 71829091413864 p^{20} T^{14} + 2508631325 p^{30} T^{16} - 63774 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
11 \( ( 1 - 699642 T^{2} + 243730562789 T^{4} - 5167862023355496 p T^{6} + \)\(10\!\cdots\!46\)\( T^{8} - \)\(17\!\cdots\!12\)\( T^{10} + \)\(10\!\cdots\!46\)\( p^{10} T^{12} - 5167862023355496 p^{21} T^{14} + 243730562789 p^{30} T^{16} - 699642 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
13 \( ( 1 - 1723346 T^{2} + 1796192221653 T^{4} - 1275220091760713112 T^{6} + \)\(69\!\cdots\!74\)\( T^{8} - \)\(28\!\cdots\!24\)\( T^{10} + \)\(69\!\cdots\!74\)\( p^{10} T^{12} - 1275220091760713112 p^{20} T^{14} + 1796192221653 p^{30} T^{16} - 1723346 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
17 \( ( 1 - 5921674 T^{2} + 16210933829709 T^{4} - 28076383539701171064 T^{6} + \)\(36\!\cdots\!46\)\( T^{8} - \)\(45\!\cdots\!60\)\( T^{10} + \)\(36\!\cdots\!46\)\( p^{10} T^{12} - 28076383539701171064 p^{20} T^{14} + 16210933829709 p^{30} T^{16} - 5921674 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
19 \( ( 1 + 11864886 T^{2} + 77718769794773 T^{4} + \)\(34\!\cdots\!92\)\( T^{6} + \)\(11\!\cdots\!82\)\( T^{8} + \)\(32\!\cdots\!32\)\( T^{10} + \)\(11\!\cdots\!82\)\( p^{10} T^{12} + \)\(34\!\cdots\!92\)\( p^{20} T^{14} + 77718769794773 p^{30} T^{16} + 11864886 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
23 \( ( 1 - 1528 T + 16750163 T^{2} - 30494836000 T^{3} + 181299677525418 T^{4} - 239369149735345360 T^{5} + 181299677525418 p^{5} T^{6} - 30494836000 p^{10} T^{7} + 16750163 p^{15} T^{8} - 1528 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
29 \( ( 1 + 108105178 T^{2} + 6217574540910645 T^{4} + \)\(24\!\cdots\!00\)\( T^{6} + \)\(72\!\cdots\!38\)\( T^{8} + \)\(16\!\cdots\!84\)\( T^{10} + \)\(72\!\cdots\!38\)\( p^{10} T^{12} + \)\(24\!\cdots\!00\)\( p^{20} T^{14} + 6217574540910645 p^{30} T^{16} + 108105178 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
31 \( ( 1 - 124893838 T^{2} + 8535334798557357 T^{4} - \)\(40\!\cdots\!44\)\( T^{6} + \)\(14\!\cdots\!34\)\( T^{8} - \)\(45\!\cdots\!12\)\( T^{10} + \)\(14\!\cdots\!34\)\( p^{10} T^{12} - \)\(40\!\cdots\!44\)\( p^{20} T^{14} + 8535334798557357 p^{30} T^{16} - 124893838 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
37 \( ( 1 - 483498626 T^{2} + 112886858733188229 T^{4} - \)\(16\!\cdots\!88\)\( T^{6} + \)\(17\!\cdots\!54\)\( T^{8} - \)\(14\!\cdots\!12\)\( T^{10} + \)\(17\!\cdots\!54\)\( p^{10} T^{12} - \)\(16\!\cdots\!88\)\( p^{20} T^{14} + 112886858733188229 p^{30} T^{16} - 483498626 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
41 \( ( 1 - 683056346 T^{2} + 238166724811925181 T^{4} - \)\(55\!\cdots\!12\)\( T^{6} + \)\(94\!\cdots\!30\)\( T^{8} - \)\(12\!\cdots\!08\)\( T^{10} + \)\(94\!\cdots\!30\)\( p^{10} T^{12} - \)\(55\!\cdots\!12\)\( p^{20} T^{14} + 238166724811925181 p^{30} T^{16} - 683056346 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
43 \( ( 1 + 8191410 p T^{2} + 125799691331078309 T^{4} + \)\(29\!\cdots\!88\)\( T^{6} + \)\(57\!\cdots\!46\)\( T^{8} + \)\(94\!\cdots\!72\)\( T^{10} + \)\(57\!\cdots\!46\)\( p^{10} T^{12} + \)\(29\!\cdots\!88\)\( p^{20} T^{14} + 125799691331078309 p^{30} T^{16} + 8191410 p^{41} T^{18} + p^{50} T^{20} )^{2} \)
47 \( ( 1 - 6528 T + 767922859 T^{2} - 120901854720 p T^{3} + 300594027559513930 T^{4} - \)\(18\!\cdots\!36\)\( T^{5} + 300594027559513930 p^{5} T^{6} - 120901854720 p^{11} T^{7} + 767922859 p^{15} T^{8} - 6528 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
53 \( ( 1 + 1787609162 T^{2} + 1368832768041552357 T^{4} + \)\(73\!\cdots\!84\)\( T^{6} + \)\(39\!\cdots\!02\)\( T^{8} + \)\(19\!\cdots\!48\)\( T^{10} + \)\(39\!\cdots\!02\)\( p^{10} T^{12} + \)\(73\!\cdots\!84\)\( p^{20} T^{14} + 1368832768041552357 p^{30} T^{16} + 1787609162 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
59 \( ( 1 - 5147616698 T^{2} + 12761348987329640709 T^{4} - \)\(20\!\cdots\!56\)\( T^{6} + \)\(22\!\cdots\!06\)\( T^{8} - \)\(18\!\cdots\!24\)\( T^{10} + \)\(22\!\cdots\!06\)\( p^{10} T^{12} - \)\(20\!\cdots\!56\)\( p^{20} T^{14} + 12761348987329640709 p^{30} T^{16} - 5147616698 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
61 \( ( 1 - 1955556146 T^{2} + 1575682574427862965 T^{4} - \)\(12\!\cdots\!08\)\( T^{6} + \)\(17\!\cdots\!70\)\( T^{8} - \)\(19\!\cdots\!60\)\( T^{10} + \)\(17\!\cdots\!70\)\( p^{10} T^{12} - \)\(12\!\cdots\!08\)\( p^{20} T^{14} + 1575682574427862965 p^{30} T^{16} - 1955556146 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
67 \( ( 1 + 7159517910 T^{2} + 342715510540507751 p T^{4} + \)\(43\!\cdots\!88\)\( T^{6} + \)\(57\!\cdots\!22\)\( T^{8} + \)\(72\!\cdots\!40\)\( T^{10} + \)\(57\!\cdots\!22\)\( p^{10} T^{12} + \)\(43\!\cdots\!88\)\( p^{20} T^{14} + 342715510540507751 p^{31} T^{16} + 7159517910 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
71 \( ( 1 + 18056 T + 6774825731 T^{2} + 85449010534112 T^{3} + 21117998926544638698 T^{4} + \)\(20\!\cdots\!24\)\( T^{5} + 21117998926544638698 p^{5} T^{6} + 85449010534112 p^{10} T^{7} + 6774825731 p^{15} T^{8} + 18056 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
73 \( ( 1 + 15314 T + 4269004917 T^{2} + 139409155522200 T^{3} + 10772946279207978354 T^{4} + \)\(37\!\cdots\!88\)\( T^{5} + 10772946279207978354 p^{5} T^{6} + 139409155522200 p^{10} T^{7} + 4269004917 p^{15} T^{8} + 15314 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
79 \( ( 1 - 21221091630 T^{2} + \)\(21\!\cdots\!81\)\( T^{4} - \)\(14\!\cdots\!56\)\( T^{6} + \)\(68\!\cdots\!50\)\( T^{8} - \)\(24\!\cdots\!92\)\( T^{10} + \)\(68\!\cdots\!50\)\( p^{10} T^{12} - \)\(14\!\cdots\!56\)\( p^{20} T^{14} + \)\(21\!\cdots\!81\)\( p^{30} T^{16} - 21221091630 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
83 \( ( 1 - 29570512106 T^{2} + \)\(41\!\cdots\!57\)\( T^{4} - \)\(37\!\cdots\!56\)\( T^{6} + \)\(23\!\cdots\!78\)\( T^{8} - \)\(10\!\cdots\!16\)\( T^{10} + \)\(23\!\cdots\!78\)\( p^{10} T^{12} - \)\(37\!\cdots\!56\)\( p^{20} T^{14} + \)\(41\!\cdots\!57\)\( p^{30} T^{16} - 29570512106 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
89 \( ( 1 - 14639611546 T^{2} + \)\(12\!\cdots\!77\)\( T^{4} - \)\(95\!\cdots\!20\)\( T^{6} + \)\(59\!\cdots\!94\)\( T^{8} - \)\(31\!\cdots\!32\)\( T^{10} + \)\(59\!\cdots\!94\)\( p^{10} T^{12} - \)\(95\!\cdots\!20\)\( p^{20} T^{14} + \)\(12\!\cdots\!77\)\( p^{30} T^{16} - 14639611546 p^{40} T^{18} + p^{50} T^{20} )^{2} \)
97 \( ( 1 - 23202 T + 14223120845 T^{2} + 159904888164072 T^{3} + \)\(18\!\cdots\!26\)\( T^{4} + \)\(14\!\cdots\!00\)\( T^{5} + \)\(18\!\cdots\!26\)\( p^{5} T^{6} + 159904888164072 p^{10} T^{7} + 14223120845 p^{15} T^{8} - 23202 p^{20} T^{9} + p^{25} T^{10} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.84687403718041399455846014228, −1.81903892565710898643682097232, −1.78230022555417223678423133839, −1.66415985665965804117540432036, −1.60041093538521950589345363598, −1.53885906212164590474221879121, −1.51499351829419733402638301527, −1.47424195506936382871063760277, −1.18507693355874128312060666372, −1.13164499588923528579915939216, −0.928015746372507847796980616538, −0.910580649343563328248446807242, −0.846579627700595925245763367581, −0.76475736366262834069398987739, −0.62710357309132161436960629217, −0.61347919064135652501018852058, −0.60913713957265843181553040461, −0.60263989787565815174328695222, −0.60068019769877229606578955246, −0.59131728471604477620162239775, −0.51221938169488612788646480889, −0.38278290375873712831285413058, −0.23726155572203822991481294890, −0.11838964934203209428105228704, −0.04774305056631611499042546569, 0.04774305056631611499042546569, 0.11838964934203209428105228704, 0.23726155572203822991481294890, 0.38278290375873712831285413058, 0.51221938169488612788646480889, 0.59131728471604477620162239775, 0.60068019769877229606578955246, 0.60263989787565815174328695222, 0.60913713957265843181553040461, 0.61347919064135652501018852058, 0.62710357309132161436960629217, 0.76475736366262834069398987739, 0.846579627700595925245763367581, 0.910580649343563328248446807242, 0.928015746372507847796980616538, 1.13164499588923528579915939216, 1.18507693355874128312060666372, 1.47424195506936382871063760277, 1.51499351829419733402638301527, 1.53885906212164590474221879121, 1.60041093538521950589345363598, 1.66415985665965804117540432036, 1.78230022555417223678423133839, 1.81903892565710898643682097232, 1.84687403718041399455846014228

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.