Properties

Label 12-50e6-1.1-c21e6-0-0
Degree $12$
Conductor $15625000000$
Sign $1$
Analytic cond. $7.44559\times 10^{12}$
Root an. cond. $11.8211$
Motivic weight $21$
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  − 3.14e6·4-s + 6.31e9·9-s − 1.55e11·11-s + 6.59e12·16-s + 4.79e13·19-s − 6.36e15·29-s − 5.90e15·31-s − 1.98e16·36-s − 3.78e17·41-s + 4.88e17·44-s + 1.57e18·49-s + 1.60e19·59-s + 5.24e18·61-s − 1.15e19·64-s + 1.01e20·71-s − 1.50e20·76-s + 4.77e20·79-s − 4.45e19·81-s + 1.29e21·89-s − 9.80e20·99-s + 2.39e21·101-s − 1.39e21·109-s + 2.00e22·116-s − 2.73e22·121-s + 1.85e22·124-s + ⋯
L(s)  = 1  − 3/2·4-s + 0.604·9-s − 1.80·11-s + 3/2·16-s + 1.79·19-s − 2.81·29-s − 1.29·31-s − 0.906·36-s − 4.40·41-s + 2.70·44-s + 2.82·49-s + 4.09·59-s + 0.941·61-s − 5/4·64-s + 3.69·71-s − 2.68·76-s + 5.67·79-s − 0.406·81-s + 4.40·89-s − 1.08·99-s + 2.15·101-s − 0.563·109-s + 4.21·116-s − 3.68·121-s + 1.93·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(22-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+21/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(12\)
Conductor: \(2^{6} \cdot 5^{12}\)
Sign: $1$
Analytic conductor: \(7.44559\times 10^{12}\)
Root analytic conductor: \(11.8211\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((12,\ 2^{6} \cdot 5^{12} ,\ ( \ : [21/2]^{6} ),\ 1 )\)

Particular Values

\(L(11)\) \(\approx\) \(9.877358452\)
\(L(\frac12)\) \(\approx\) \(9.877358452\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{20} T^{2} )^{3} \)
5 \( 1 \)
good3 \( 1 - 702024695 p^{2} T^{2} + 12869855025400582 p^{8} T^{4} - \)\(49\!\cdots\!15\)\( p^{16} T^{6} + 12869855025400582 p^{50} T^{8} - 702024695 p^{86} T^{10} + p^{126} T^{12} \)
7 \( 1 - 1575890680050164190 T^{2} + \)\(18\!\cdots\!03\)\( p^{2} T^{4} - \)\(15\!\cdots\!20\)\( p^{4} T^{6} + \)\(18\!\cdots\!03\)\( p^{44} T^{8} - 1575890680050164190 p^{84} T^{10} + p^{126} T^{12} \)
11 \( ( 1 + 77565926349 T + \)\(20\!\cdots\!00\)\( p T^{2} + \)\(94\!\cdots\!65\)\( p^{2} T^{3} + \)\(20\!\cdots\!00\)\( p^{22} T^{4} + 77565926349 p^{42} T^{5} + p^{63} T^{6} )^{2} \)
13 \( 1 - \)\(12\!\cdots\!10\)\( T^{2} + \)\(66\!\cdots\!07\)\( T^{4} - \)\(12\!\cdots\!20\)\( p^{2} T^{6} + \)\(66\!\cdots\!07\)\( p^{42} T^{8} - \)\(12\!\cdots\!10\)\( p^{84} T^{10} + p^{126} T^{12} \)
17 \( 1 - \)\(73\!\cdots\!35\)\( p^{2} T^{2} + \)\(31\!\cdots\!02\)\( p^{4} T^{4} - \)\(91\!\cdots\!55\)\( p^{6} T^{6} + \)\(31\!\cdots\!02\)\( p^{46} T^{8} - \)\(73\!\cdots\!35\)\( p^{86} T^{10} + p^{126} T^{12} \)
19 \( ( 1 - 1261032950775 p T + \)\(48\!\cdots\!12\)\( p^{2} T^{2} - \)\(36\!\cdots\!75\)\( p^{3} T^{3} + \)\(48\!\cdots\!12\)\( p^{23} T^{4} - 1261032950775 p^{43} T^{5} + p^{63} T^{6} )^{2} \)
23 \( 1 - \)\(10\!\cdots\!70\)\( T^{2} + \)\(53\!\cdots\!87\)\( T^{4} - \)\(21\!\cdots\!60\)\( T^{6} + \)\(53\!\cdots\!87\)\( p^{42} T^{8} - \)\(10\!\cdots\!70\)\( p^{84} T^{10} + p^{126} T^{12} \)
29 \( ( 1 + 3183619096555920 T + \)\(83\!\cdots\!87\)\( T^{2} + \)\(13\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!87\)\( p^{21} T^{4} + 3183619096555920 p^{42} T^{5} + p^{63} T^{6} )^{2} \)
31 \( ( 1 + 2950614234339474 T + \)\(23\!\cdots\!85\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(23\!\cdots\!85\)\( p^{21} T^{4} + 2950614234339474 p^{42} T^{5} + p^{63} T^{6} )^{2} \)
37 \( 1 - \)\(46\!\cdots\!50\)\( T^{2} + \)\(93\!\cdots\!07\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(93\!\cdots\!07\)\( p^{42} T^{8} - \)\(46\!\cdots\!50\)\( p^{84} T^{10} + p^{126} T^{12} \)
41 \( ( 1 + 189362455401586329 T + \)\(28\!\cdots\!70\)\( T^{2} + \)\(25\!\cdots\!85\)\( T^{3} + \)\(28\!\cdots\!70\)\( p^{21} T^{4} + 189362455401586329 p^{42} T^{5} + p^{63} T^{6} )^{2} \)
43 \( 1 - \)\(99\!\cdots\!30\)\( T^{2} + \)\(44\!\cdots\!47\)\( T^{4} - \)\(11\!\cdots\!40\)\( T^{6} + \)\(44\!\cdots\!47\)\( p^{42} T^{8} - \)\(99\!\cdots\!30\)\( p^{84} T^{10} + p^{126} T^{12} \)
47 \( 1 - \)\(43\!\cdots\!10\)\( T^{2} + \)\(28\!\cdots\!27\)\( T^{4} - \)\(21\!\cdots\!80\)\( T^{6} + \)\(28\!\cdots\!27\)\( p^{42} T^{8} - \)\(43\!\cdots\!10\)\( p^{84} T^{10} + p^{126} T^{12} \)
53 \( 1 - \)\(79\!\cdots\!90\)\( T^{2} + \)\(28\!\cdots\!27\)\( T^{4} - \)\(58\!\cdots\!20\)\( T^{6} + \)\(28\!\cdots\!27\)\( p^{42} T^{8} - \)\(79\!\cdots\!90\)\( p^{84} T^{10} + p^{126} T^{12} \)
59 \( ( 1 - 8039229184581194160 T + \)\(37\!\cdots\!77\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!77\)\( p^{21} T^{4} - 8039229184581194160 p^{42} T^{5} + p^{63} T^{6} )^{2} \)
61 \( ( 1 - 2623888488791543046 T + \)\(37\!\cdots\!55\)\( T^{2} - \)\(19\!\cdots\!80\)\( T^{3} + \)\(37\!\cdots\!55\)\( p^{21} T^{4} - 2623888488791543046 p^{42} T^{5} + p^{63} T^{6} )^{2} \)
67 \( 1 - \)\(63\!\cdots\!75\)\( T^{2} + \)\(22\!\cdots\!42\)\( T^{4} - \)\(57\!\cdots\!75\)\( T^{6} + \)\(22\!\cdots\!42\)\( p^{42} T^{8} - \)\(63\!\cdots\!75\)\( p^{84} T^{10} + p^{126} T^{12} \)
71 \( ( 1 - 50615572234903950276 T + \)\(28\!\cdots\!05\)\( T^{2} - \)\(77\!\cdots\!80\)\( T^{3} + \)\(28\!\cdots\!05\)\( p^{21} T^{4} - 50615572234903950276 p^{42} T^{5} + p^{63} T^{6} )^{2} \)
73 \( 1 - \)\(28\!\cdots\!55\)\( T^{2} + \)\(80\!\cdots\!62\)\( T^{4} - \)\(11\!\cdots\!15\)\( T^{6} + \)\(80\!\cdots\!62\)\( p^{42} T^{8} - \)\(28\!\cdots\!55\)\( p^{84} T^{10} + p^{126} T^{12} \)
79 \( ( 1 - \)\(23\!\cdots\!70\)\( T + \)\(39\!\cdots\!37\)\( T^{2} - \)\(38\!\cdots\!60\)\( T^{3} + \)\(39\!\cdots\!37\)\( p^{21} T^{4} - \)\(23\!\cdots\!70\)\( p^{42} T^{5} + p^{63} T^{6} )^{2} \)
83 \( 1 - \)\(94\!\cdots\!55\)\( T^{2} + \)\(40\!\cdots\!42\)\( T^{4} - \)\(10\!\cdots\!15\)\( T^{6} + \)\(40\!\cdots\!42\)\( p^{42} T^{8} - \)\(94\!\cdots\!55\)\( p^{84} T^{10} + p^{126} T^{12} \)
89 \( ( 1 - \)\(64\!\cdots\!55\)\( T + \)\(35\!\cdots\!42\)\( T^{2} - \)\(11\!\cdots\!15\)\( T^{3} + \)\(35\!\cdots\!42\)\( p^{21} T^{4} - \)\(64\!\cdots\!55\)\( p^{42} T^{5} + p^{63} T^{6} )^{2} \)
97 \( 1 - \)\(13\!\cdots\!70\)\( T^{2} + \)\(86\!\cdots\!27\)\( T^{4} - \)\(47\!\cdots\!60\)\( T^{6} + \)\(86\!\cdots\!27\)\( p^{42} T^{8} - \)\(13\!\cdots\!70\)\( p^{84} T^{10} + p^{126} T^{12} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.08283600589039106777926751039, −5.02834757518804342221631973943, −4.92681030021659894287104589824, −4.80720988044480968601870222249, −4.02485580417996297048365676686, −3.80840304005502823770604460927, −3.80610958479561450419159157655, −3.75465655855818433421290412092, −3.70573830806811808207781921006, −3.58639329643724413087734174080, −3.06416942654994681796095911359, −2.75854892864085588712123007344, −2.74798979927742836544401875214, −2.33837254708280601196728416524, −2.19294184349310745229076614890, −2.00137008357279076273938992959, −1.77896898558320070385124674674, −1.63369194100483041403941220081, −1.48516634045337926359287314511, −0.963860607192440324154422274797, −0.78160322340820774005063530781, −0.57626109160075253622732588977, −0.55118660905697528899142308427, −0.44827893462329754674462908162, −0.27184188910551005625731584678, 0.27184188910551005625731584678, 0.44827893462329754674462908162, 0.55118660905697528899142308427, 0.57626109160075253622732588977, 0.78160322340820774005063530781, 0.963860607192440324154422274797, 1.48516634045337926359287314511, 1.63369194100483041403941220081, 1.77896898558320070385124674674, 2.00137008357279076273938992959, 2.19294184349310745229076614890, 2.33837254708280601196728416524, 2.74798979927742836544401875214, 2.75854892864085588712123007344, 3.06416942654994681796095911359, 3.58639329643724413087734174080, 3.70573830806811808207781921006, 3.75465655855818433421290412092, 3.80610958479561450419159157655, 3.80840304005502823770604460927, 4.02485580417996297048365676686, 4.80720988044480968601870222249, 4.92681030021659894287104589824, 5.02834757518804342221631973943, 5.08283600589039106777926751039

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

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