# Properties

 Label 4-39e4-1.1-c3e2-0-5 Degree $4$ Conductor $2313441$ Sign $1$ Analytic cond. $8053.60$ Root an. cond. $9.47322$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $2$

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## Dirichlet series

 L(s)  = 1 − 16·4-s + 192·16-s − 250·25-s + 1.04e3·43-s + 286·49-s − 364·61-s − 2.04e3·64-s − 1.76e3·79-s + 4.00e3·100-s − 3.64e3·103-s − 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.66e4·172-s + 173-s + 179-s + 181-s + 191-s + 193-s − 4.57e3·196-s + ⋯
 L(s)  = 1 − 2·4-s + 3·16-s − 2·25-s + 3.68·43-s + 0.833·49-s − 0.764·61-s − 4·64-s − 2.51·79-s + 4·100-s − 3.48·103-s − 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 7.37·172-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s − 1.66·196-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$2313441$$    =    $$3^{4} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$8053.60$$ Root analytic conductor: $$9.47322$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{1521} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$2$$ Selberg data: $$(4,\ 2313441,\ (\ :3/2, 3/2),\ 1)$$

## Particular Values

 $$L(2)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{5}{2})$$ 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)$
bad3 $$1$$
13 $$1$$
good2$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
5$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
7$C_2^2$ $$1 - 286 T^{2} + p^{6} T^{4}$$
11$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
17$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
19$C_2^2$ $$1 - 10582 T^{2} + p^{6} T^{4}$$
23$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
29$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
31$C_2^2$ $$1 + 35282 T^{2} + p^{6} T^{4}$$
37$C_2^2$ $$1 - 89206 T^{2} + p^{6} T^{4}$$
41$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
43$C_2$ $$( 1 - 520 T + p^{3} T^{2} )^{2}$$
47$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
53$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
59$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
61$C_2$ $$( 1 + 182 T + p^{3} T^{2} )^{2}$$
67$C_2^2$ $$1 + 172874 T^{2} + p^{6} T^{4}$$
71$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
73$C_2^2$ $$1 + 638066 T^{2} + p^{6} T^{4}$$
79$C_2$ $$( 1 + 884 T + p^{3} T^{2} )^{2}$$
83$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
89$C_2$ $$( 1 + p^{3} T^{2} )^{2}$$
97$C_2^2$ $$1 - 56446 T^{2} + p^{6} T^{4}$$
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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

−8.998214003713661364437147075987, −8.490328606881175717517447574420, −8.216853897304774644156514369066, −7.83327389115685696044828067506, −7.28937037436413480601284473953, −7.21079640178592173931393431180, −6.06808775661276535288782590172, −6.01113100590954679548542440306, −5.57895385905292561629522994632, −5.21533422105478407613540508452, −4.47156567398099798176236753724, −4.33717937869256377580484571026, −3.89040580809538059225491106695, −3.55505660731407155280105406811, −2.77187807113650451896034189970, −2.30600523739104581527236488882, −1.31732405888759754477880926371, −1.00655724279934310474327491070, 0, 0, 1.00655724279934310474327491070, 1.31732405888759754477880926371, 2.30600523739104581527236488882, 2.77187807113650451896034189970, 3.55505660731407155280105406811, 3.89040580809538059225491106695, 4.33717937869256377580484571026, 4.47156567398099798176236753724, 5.21533422105478407613540508452, 5.57895385905292561629522994632, 6.01113100590954679548542440306, 6.06808775661276535288782590172, 7.21079640178592173931393431180, 7.28937037436413480601284473953, 7.83327389115685696044828067506, 8.216853897304774644156514369066, 8.490328606881175717517447574420, 8.998214003713661364437147075987