Properties

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

Origins of factors

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$D_{4}$ $$1 -$$$$60\!\cdots\!00$$$$T +$$$$57\!\cdots\!70$$$$T^{2} -$$$$60\!\cdots\!00$$$$p^{37} T^{3} + p^{74} T^{4}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

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