Properties

Degree $4$
Conductor $1$
Sign $1$
Motivic weight $37$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.94e5·2-s + 1.39e7·3-s − 9.96e10·4-s + 5.52e12·5-s − 2.71e12·6-s − 3.44e15·7-s + 1.93e16·8-s − 8.99e17·9-s − 1.07e18·10-s − 2.67e19·11-s − 1.39e18·12-s + 5.30e20·13-s + 6.70e20·14-s + 7.73e19·15-s − 3.76e21·16-s − 8.94e22·17-s + 1.74e23·18-s + 3.73e23·19-s − 5.51e23·20-s − 4.82e22·21-s + 5.19e24·22-s − 2.62e25·23-s + 2.71e23·24-s − 2.20e23·25-s − 1.03e26·26-s − 1.88e25·27-s + 3.43e26·28-s + ⋯
L(s)  = 1  − 0.524·2-s + 0.0208·3-s − 0.725·4-s + 0.648·5-s − 0.0109·6-s − 0.800·7-s + 0.380·8-s − 1.99·9-s − 0.339·10-s − 1.44·11-s − 0.0151·12-s + 1.30·13-s + 0.419·14-s + 0.0135·15-s − 0.199·16-s − 1.54·17-s + 1.04·18-s + 0.822·19-s − 0.470·20-s − 0.0166·21-s + 0.760·22-s − 1.68·23-s + 0.00793·24-s − 0.00302·25-s − 0.686·26-s − 0.0624·27-s + 0.580·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(38-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+37/2)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1\)
Sign: $1$
Motivic weight: \(37\)
Character: $\chi_{1} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1,\ (\ :37/2, 37/2),\ 1)\)

Particular Values

\(L(19)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{39}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
good2$D_{4}$ \( 1 + 6075 p^{5} T + 16781555 p^{13} T^{2} + 6075 p^{42} T^{3} + p^{74} T^{4} \)
3$D_{4}$ \( 1 - 518200 p^{3} T + 15239131479430 p^{10} T^{2} - 518200 p^{40} T^{3} + p^{74} T^{4} \)
5$D_{4}$ \( 1 - 221183375436 p^{2} T + \)\(39\!\cdots\!58\)\( p^{7} T^{2} - 221183375436 p^{39} T^{3} + p^{74} T^{4} \)
7$D_{4}$ \( 1 + 70376407214000 p^{2} T + \)\(15\!\cdots\!50\)\( p^{5} T^{2} + 70376407214000 p^{39} T^{3} + p^{74} T^{4} \)
11$D_{4}$ \( 1 + 2430366941349867096 p T + \)\(53\!\cdots\!46\)\( p^{3} T^{2} + 2430366941349867096 p^{38} T^{3} + p^{74} T^{4} \)
13$D_{4}$ \( 1 - 40813980127225629100 p T + \)\(14\!\cdots\!70\)\( p^{3} T^{2} - 40813980127225629100 p^{38} T^{3} + p^{74} T^{4} \)
17$D_{4}$ \( 1 + \)\(52\!\cdots\!00\)\( p T + \)\(17\!\cdots\!10\)\( p^{3} T^{2} + \)\(52\!\cdots\!00\)\( p^{38} T^{3} + p^{74} T^{4} \)
19$D_{4}$ \( 1 - \)\(37\!\cdots\!20\)\( T + \)\(10\!\cdots\!62\)\( p T^{2} - \)\(37\!\cdots\!20\)\( p^{37} T^{3} + p^{74} T^{4} \)
23$D_{4}$ \( 1 + \)\(26\!\cdots\!00\)\( T + \)\(21\!\cdots\!70\)\( p T^{2} + \)\(26\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \)
29$D_{4}$ \( 1 + \)\(43\!\cdots\!80\)\( p T + \)\(26\!\cdots\!98\)\( p^{2} T^{2} + \)\(43\!\cdots\!80\)\( p^{38} T^{3} + p^{74} T^{4} \)
31$D_{4}$ \( 1 - \)\(84\!\cdots\!04\)\( p T + \)\(25\!\cdots\!06\)\( p^{2} T^{2} - \)\(84\!\cdots\!04\)\( p^{38} T^{3} + p^{74} T^{4} \)
37$D_{4}$ \( 1 + \)\(49\!\cdots\!00\)\( p^{2} T + \)\(13\!\cdots\!90\)\( T^{2} + \)\(49\!\cdots\!00\)\( p^{39} T^{3} + p^{74} T^{4} \)
41$D_{4}$ \( 1 + \)\(12\!\cdots\!36\)\( T + \)\(12\!\cdots\!86\)\( T^{2} + \)\(12\!\cdots\!36\)\( p^{37} T^{3} + p^{74} T^{4} \)
43$D_{4}$ \( 1 + \)\(25\!\cdots\!00\)\( T + \)\(44\!\cdots\!50\)\( T^{2} + \)\(25\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \)
47$D_{4}$ \( 1 - \)\(42\!\cdots\!00\)\( T + \)\(14\!\cdots\!70\)\( T^{2} - \)\(42\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \)
53$D_{4}$ \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(16\!\cdots\!70\)\( T^{2} - \)\(15\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \)
59$D_{4}$ \( 1 + \)\(23\!\cdots\!40\)\( T + \)\(64\!\cdots\!38\)\( T^{2} + \)\(23\!\cdots\!40\)\( p^{37} T^{3} + p^{74} T^{4} \)
61$D_{4}$ \( 1 - \)\(10\!\cdots\!44\)\( T + \)\(10\!\cdots\!26\)\( T^{2} - \)\(10\!\cdots\!44\)\( p^{37} T^{3} + p^{74} T^{4} \)
67$D_{4}$ \( 1 + \)\(10\!\cdots\!00\)\( T + \)\(94\!\cdots\!30\)\( T^{2} + \)\(10\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \)
71$D_{4}$ \( 1 + \)\(74\!\cdots\!16\)\( T + \)\(58\!\cdots\!46\)\( T^{2} + \)\(74\!\cdots\!16\)\( p^{37} T^{3} + p^{74} T^{4} \)
73$D_{4}$ \( 1 - \)\(19\!\cdots\!00\)\( T + \)\(18\!\cdots\!10\)\( T^{2} - \)\(19\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \)
79$D_{4}$ \( 1 - \)\(27\!\cdots\!80\)\( T + \)\(64\!\cdots\!42\)\( p T^{2} - \)\(27\!\cdots\!80\)\( p^{37} T^{3} + p^{74} T^{4} \)
83$D_{4}$ \( 1 + \)\(47\!\cdots\!00\)\( T + \)\(24\!\cdots\!30\)\( T^{2} + \)\(47\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \)
89$D_{4}$ \( 1 + \)\(13\!\cdots\!60\)\( T + \)\(23\!\cdots\!58\)\( T^{2} + \)\(13\!\cdots\!60\)\( p^{37} T^{3} + p^{74} T^{4} \)
97$D_{4}$ \( 1 - \)\(60\!\cdots\!00\)\( T + \)\(57\!\cdots\!70\)\( T^{2} - \)\(60\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.76929614383051956740114757090, −22.23571281374329264203523647512, −20.63653358861324988635101867697, −19.87494481669384388760742054607, −18.24871950005246888891702940735, −18.00274185120459235642841690396, −16.71303239842308479600166267384, −15.51415077536740871668288185592, −13.58579983358184777956746156554, −13.54710833338065349296574072216, −11.50132024290830732411905645660, −10.20458218628807267645251886245, −8.972504432680970898032209122777, −8.313372276419025684925283884069, −6.23021964751350246881882725271, −5.30469258083104533062022796873, −3.40942703835798794686937639842, −2.19691573656503192005496061157, 0, 0, 2.19691573656503192005496061157, 3.40942703835798794686937639842, 5.30469258083104533062022796873, 6.23021964751350246881882725271, 8.313372276419025684925283884069, 8.972504432680970898032209122777, 10.20458218628807267645251886245, 11.50132024290830732411905645660, 13.54710833338065349296574072216, 13.58579983358184777956746156554, 15.51415077536740871668288185592, 16.71303239842308479600166267384, 18.00274185120459235642841690396, 18.24871950005246888891702940735, 19.87494481669384388760742054607, 20.63653358861324988635101867697, 22.23571281374329264203523647512, 22.76929614383051956740114757090

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