Properties

Label 24-384e12-1.1-c7e12-0-0
Degree $24$
Conductor $1.028\times 10^{31}$
Sign $1$
Analytic cond. $8.87681\times 10^{24}$
Root an. cond. $10.9524$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4.37e3·9-s − 8.30e4·17-s + 4.04e5·25-s − 9.69e5·41-s − 7.48e6·49-s + 1.80e7·73-s + 1.11e7·81-s + 4.35e7·89-s − 4.84e7·97-s − 8.76e7·113-s + 1.52e8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 3.63e8·153-s + 157-s + 163-s + 167-s + 4.45e8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 2·9-s − 4.10·17-s + 5.18·25-s − 2.19·41-s − 9.08·49-s + 5.43·73-s + 7/3·81-s + 6.55·89-s − 5.39·97-s − 5.71·113-s + 7.80·121-s + 8.20·153-s + 7.10·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{84} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(24\)
Conductor: \(2^{84} \cdot 3^{12}\)
Sign: $1$
Analytic conductor: \(8.87681\times 10^{24}\)
Root analytic conductor: \(10.9524\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{384} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((24,\ 2^{84} \cdot 3^{12} ,\ ( \ : [7/2]^{12} ),\ 1 )\)

Particular Values

\(L(4)\) \(\approx\) \(0.7955399625\)
\(L(\frac12)\) \(\approx\) \(0.7955399625\)
\(L(\frac{9}{2})\) 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 \( ( 1 + p^{6} T^{2} )^{6} \)
good5 \( ( 1 - 202422 T^{2} + 150456147 p^{3} T^{4} - 2272723482452 p^{4} T^{6} + 150456147 p^{17} T^{8} - 202422 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
7 \( ( 1 + 3742386 T^{2} + 6535894742751 T^{4} + 6805623648777266172 T^{6} + 6535894742751 p^{14} T^{8} + 3742386 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
11 \( ( 1 - 76035810 T^{2} + 2907631859152071 T^{4} - \)\(69\!\cdots\!84\)\( T^{6} + 2907631859152071 p^{14} T^{8} - 76035810 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
13 \( ( 1 - 222990078 T^{2} + 20968007553597495 T^{4} - \)\(13\!\cdots\!60\)\( T^{6} + 20968007553597495 p^{14} T^{8} - 222990078 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
17 \( ( 1 + 1222 p T + 710809823 T^{2} + 17058319559700 T^{3} + 710809823 p^{7} T^{4} + 1222 p^{15} T^{5} + p^{21} T^{6} )^{4} \)
19 \( ( 1 - 1924649010 T^{2} + 2618941304588675895 T^{4} - \)\(26\!\cdots\!52\)\( T^{6} + 2618941304588675895 p^{14} T^{8} - 1924649010 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
23 \( ( 1 + 8324910090 T^{2} + 43626867506508510495 T^{4} + \)\(16\!\cdots\!04\)\( T^{6} + 43626867506508510495 p^{14} T^{8} + 8324910090 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
29 \( ( 1 - 74216131110 T^{2} + \)\(26\!\cdots\!43\)\( T^{4} - \)\(58\!\cdots\!20\)\( T^{6} + \)\(26\!\cdots\!43\)\( p^{14} T^{8} - 74216131110 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
31 \( ( 1 + 92397120738 T^{2} + \)\(40\!\cdots\!19\)\( T^{4} + \)\(12\!\cdots\!76\)\( T^{6} + \)\(40\!\cdots\!19\)\( p^{14} T^{8} + 92397120738 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
37 \( ( 1 - 301868135502 T^{2} + \)\(54\!\cdots\!67\)\( T^{4} - \)\(61\!\cdots\!40\)\( T^{6} + \)\(54\!\cdots\!67\)\( p^{14} T^{8} - 301868135502 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
41 \( ( 1 + 242258 T + 157515778679 T^{2} + 120350837328774876 T^{3} + 157515778679 p^{7} T^{4} + 242258 p^{14} T^{5} + p^{21} T^{6} )^{4} \)
43 \( ( 1 - 1139303049378 T^{2} + \)\(57\!\cdots\!27\)\( T^{4} - \)\(18\!\cdots\!44\)\( T^{6} + \)\(57\!\cdots\!27\)\( p^{14} T^{8} - 1139303049378 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
47 \( ( 1 + 2213950484346 T^{2} + \)\(22\!\cdots\!87\)\( T^{4} + \)\(13\!\cdots\!96\)\( T^{6} + \)\(22\!\cdots\!87\)\( p^{14} T^{8} + 2213950484346 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
53 \( ( 1 - 4140796683414 T^{2} + \)\(98\!\cdots\!47\)\( T^{4} - \)\(14\!\cdots\!84\)\( T^{6} + \)\(98\!\cdots\!47\)\( p^{14} T^{8} - 4140796683414 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
59 \( ( 1 - 9172921230786 T^{2} + \)\(45\!\cdots\!15\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(45\!\cdots\!15\)\( p^{14} T^{8} - 9172921230786 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
61 \( ( 1 - 11980240333758 T^{2} + \)\(75\!\cdots\!03\)\( T^{4} - \)\(29\!\cdots\!92\)\( T^{6} + \)\(75\!\cdots\!03\)\( p^{14} T^{8} - 11980240333758 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
67 \( ( 1 - 25642072420626 T^{2} + \)\(47\!\cdots\!37\)\( p T^{4} - \)\(24\!\cdots\!96\)\( T^{6} + \)\(47\!\cdots\!37\)\( p^{15} T^{8} - 25642072420626 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
71 \( ( 1 + 5015239048170 T^{2} + \)\(99\!\cdots\!43\)\( T^{4} + \)\(12\!\cdots\!40\)\( T^{6} + \)\(99\!\cdots\!43\)\( p^{14} T^{8} + 5015239048170 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
73 \( ( 1 - 4519914 T + 9068728164087 T^{2} - 3826246015633904620 T^{3} + 9068728164087 p^{7} T^{4} - 4519914 p^{14} T^{5} + p^{21} T^{6} )^{4} \)
79 \( ( 1 + 69274085093826 T^{2} + \)\(26\!\cdots\!03\)\( T^{4} + \)\(61\!\cdots\!12\)\( T^{6} + \)\(26\!\cdots\!03\)\( p^{14} T^{8} + 69274085093826 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
83 \( ( 1 - 40550002231986 T^{2} + \)\(18\!\cdots\!91\)\( T^{4} - \)\(50\!\cdots\!28\)\( T^{6} + \)\(18\!\cdots\!91\)\( p^{14} T^{8} - 40550002231986 p^{28} T^{10} + p^{42} T^{12} )^{2} \)
89 \( ( 1 - 10890786 T + 89854806381351 T^{2} - \)\(74\!\cdots\!92\)\( T^{3} + 89854806381351 p^{7} T^{4} - 10890786 p^{14} T^{5} + p^{21} T^{6} )^{4} \)
97 \( ( 1 + 12114582 T + 207440378530095 T^{2} + \)\(17\!\cdots\!24\)\( T^{3} + 207440378530095 p^{7} T^{4} + 12114582 p^{14} T^{5} + p^{21} T^{6} )^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.67257397961257434725082390762, −2.43652014409741548194612165022, −2.34574337203488733611275729664, −2.33375438195196182201754403449, −2.10265879030147209007247627306, −2.08996231882497755231216559658, −1.92799944693497375714297369255, −1.92587144695946162425599078921, −1.81871405641475249808268924049, −1.73345064028086736593425648599, −1.67702476505267422827897945120, −1.49511638232391986078750934596, −1.46194695971321191271856057459, −1.20765530163093543598419337961, −0.968217088121825989258314320937, −0.955094028623138771518194415284, −0.903353429551993335403204577830, −0.854642493021223640373072558724, −0.77717204180599831872126024118, −0.48826585568928068247654790016, −0.42331050621680193658613247190, −0.38071144948694189106907705665, −0.26387527481434694970922160138, −0.11051996825247442580790560839, −0.07381753002903350747343313731, 0.07381753002903350747343313731, 0.11051996825247442580790560839, 0.26387527481434694970922160138, 0.38071144948694189106907705665, 0.42331050621680193658613247190, 0.48826585568928068247654790016, 0.77717204180599831872126024118, 0.854642493021223640373072558724, 0.903353429551993335403204577830, 0.955094028623138771518194415284, 0.968217088121825989258314320937, 1.20765530163093543598419337961, 1.46194695971321191271856057459, 1.49511638232391986078750934596, 1.67702476505267422827897945120, 1.73345064028086736593425648599, 1.81871405641475249808268924049, 1.92587144695946162425599078921, 1.92799944693497375714297369255, 2.08996231882497755231216559658, 2.10265879030147209007247627306, 2.33375438195196182201754403449, 2.34574337203488733611275729664, 2.43652014409741548194612165022, 2.67257397961257434725082390762

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

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