# Properties

 Label 4-7e4-1.1-c5e2-0-6 Degree $4$ Conductor $2401$ Sign $1$ Analytic cond. $61.7608$ Root an. cond. $2.80335$ Motivic weight $5$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·2-s − 8·3-s − 24·4-s − 38·5-s − 16·6-s − 40·8-s − 401·9-s − 76·10-s + 424·11-s + 192·12-s − 924·13-s + 304·15-s − 304·16-s − 2.34e3·17-s − 802·18-s − 360·19-s + 912·20-s + 848·22-s − 12·23-s + 320·24-s − 1.46e3·25-s − 1.84e3·26-s + 4.98e3·27-s − 7.05e3·29-s + 608·30-s + 3.54e3·31-s − 3.07e3·32-s + ⋯
 L(s)  = 1 + 0.353·2-s − 0.513·3-s − 3/4·4-s − 0.679·5-s − 0.181·6-s − 0.220·8-s − 1.65·9-s − 0.240·10-s + 1.05·11-s + 0.384·12-s − 1.51·13-s + 0.348·15-s − 0.296·16-s − 1.96·17-s − 0.583·18-s − 0.228·19-s + 0.509·20-s + 0.373·22-s − 0.00473·23-s + 0.113·24-s − 0.469·25-s − 0.536·26-s + 1.31·27-s − 1.55·29-s + 0.123·30-s + 0.663·31-s − 0.530·32-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2401$$    =    $$7^{4}$$ Sign: $1$ Analytic conductor: $$61.7608$$ Root analytic conductor: $$2.80335$$ Motivic weight: $$5$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 2401,\ (\ :5/2, 5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{7}{2})$$ 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)$
bad7 $$1$$
good2$D_{4}$ $$1 - p T + 7 p^{2} T^{2} - p^{6} T^{3} + p^{10} T^{4}$$
3$D_{4}$ $$1 + 8 T + 155 p T^{2} + 8 p^{5} T^{3} + p^{10} T^{4}$$
5$D_{4}$ $$1 + 38 T + 2911 T^{2} + 38 p^{5} T^{3} + p^{10} T^{4}$$
11$D_{4}$ $$1 - 424 T + 347473 T^{2} - 424 p^{5} T^{3} + p^{10} T^{4}$$
13$D_{4}$ $$1 + 924 T + 927022 T^{2} + 924 p^{5} T^{3} + p^{10} T^{4}$$
17$D_{4}$ $$1 + 138 p T + 3570955 T^{2} + 138 p^{6} T^{3} + p^{10} T^{4}$$
19$D_{4}$ $$1 + 360 T + 2145625 T^{2} + 360 p^{5} T^{3} + p^{10} T^{4}$$
23$D_{4}$ $$1 + 12 T + 12696565 T^{2} + 12 p^{5} T^{3} + p^{10} T^{4}$$
29$D_{4}$ $$1 + 7052 T + 35324974 T^{2} + 7052 p^{5} T^{3} + p^{10} T^{4}$$
31$D_{4}$ $$1 - 3548 T + 41490053 T^{2} - 3548 p^{5} T^{3} + p^{10} T^{4}$$
37$D_{4}$ $$1 - 11090 T + 146343239 T^{2} - 11090 p^{5} T^{3} + p^{10} T^{4}$$
41$D_{4}$ $$1 - 3500 T + 206898214 T^{2} - 3500 p^{5} T^{3} + p^{10} T^{4}$$
43$D_{4}$ $$1 + 12680 T + 267378054 T^{2} + 12680 p^{5} T^{3} + p^{10} T^{4}$$
47$D_{4}$ $$1 + 22956 T + 491525173 T^{2} + 22956 p^{5} T^{3} + p^{10} T^{4}$$
53$D_{4}$ $$1 - 3042 T + 716414839 T^{2} - 3042 p^{5} T^{3} + p^{10} T^{4}$$
59$D_{4}$ $$1 + 65808 T + 2502089257 T^{2} + 65808 p^{5} T^{3} + p^{10} T^{4}$$
61$D_{4}$ $$1 + 42486 T + 1501996159 T^{2} + 42486 p^{5} T^{3} + p^{10} T^{4}$$
67$D_{4}$ $$1 - 42312 T + 3116577793 T^{2} - 42312 p^{5} T^{3} + p^{10} T^{4}$$
71$D_{4}$ $$1 + 2208 T + 3433192846 T^{2} + 2208 p^{5} T^{3} + p^{10} T^{4}$$
73$D_{4}$ $$1 + 50506 T + 2773040987 T^{2} + 50506 p^{5} T^{3} + p^{10} T^{4}$$
79$D_{4}$ $$1 - 9004 T + 5176592589 T^{2} - 9004 p^{5} T^{3} + p^{10} T^{4}$$
83$D_{4}$ $$1 + 104328 T + 9837878230 T^{2} + 104328 p^{5} T^{3} + p^{10} T^{4}$$
89$D_{4}$ $$1 + 26666 T + 8107102555 T^{2} + 26666 p^{5} T^{3} + p^{10} T^{4}$$
97$D_{4}$ $$1 - 2156 p T + 28107307478 T^{2} - 2156 p^{6} T^{3} + p^{10} T^{4}$$
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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

−14.14430956103366514539280512831, −13.97707505916677750572864290375, −13.07045261790796160774708673764, −12.67105801810661136471000522684, −11.76842490935789868639683464795, −11.37549949434715106644384913737, −11.26116481820268018581504288932, −10.12561173926461168098254101013, −9.135193729930420440753916975988, −9.071531298190257858435339852059, −8.180280829456720905351442699079, −7.39900359603591289223417542366, −6.47594150688748095783940565143, −5.90616051394942511995163511068, −4.70786807799000360944408023498, −4.56893824872503263530190255472, −3.38413568468120653904066548416, −2.21371360549040054329945564287, 0, 0, 2.21371360549040054329945564287, 3.38413568468120653904066548416, 4.56893824872503263530190255472, 4.70786807799000360944408023498, 5.90616051394942511995163511068, 6.47594150688748095783940565143, 7.39900359603591289223417542366, 8.180280829456720905351442699079, 9.071531298190257858435339852059, 9.135193729930420440753916975988, 10.12561173926461168098254101013, 11.26116481820268018581504288932, 11.37549949434715106644384913737, 11.76842490935789868639683464795, 12.67105801810661136471000522684, 13.07045261790796160774708673764, 13.97707505916677750572864290375, 14.14430956103366514539280512831