# Properties

 Degree 4 Conductor $2^{8} \cdot 3^{4} \cdot 7^{2}$ Sign $1$ Motivic weight 1 Primitive no Self-dual yes Analytic rank 0

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 5-s + 7-s − 3·9-s − 6·11-s − 6·13-s − 4·17-s − 14·19-s − 23-s + 5·25-s − 2·29-s + 10·31-s + 35-s − 12·37-s + 8·41-s − 10·43-s − 3·45-s + 8·47-s + 4·53-s − 6·55-s − 7·61-s − 3·63-s − 6·65-s − 12·67-s − 30·71-s − 4·73-s − 6·77-s + 79-s + ⋯
 L(s)  = 1 + 0.447·5-s + 0.377·7-s − 9-s − 1.80·11-s − 1.66·13-s − 0.970·17-s − 3.21·19-s − 0.208·23-s + 25-s − 0.371·29-s + 1.79·31-s + 0.169·35-s − 1.97·37-s + 1.24·41-s − 1.52·43-s − 0.447·45-s + 1.16·47-s + 0.549·53-s − 0.809·55-s − 0.896·61-s − 0.377·63-s − 0.744·65-s − 1.46·67-s − 3.56·71-s − 0.468·73-s − 0.683·77-s + 0.112·79-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$4$$ $$N$$ = $$1016064$$    =    $$2^{8} \cdot 3^{4} \cdot 7^{2}$$ $$\varepsilon$$ = $1$ motivic weight = $$1$$ character : induced by $\chi_{1008} (1, \cdot )$ primitive : no self-dual : yes analytic rank = $$0$$ Selberg data = $$(4,\ 1016064,\ (\ :1/2, 1/2),\ 1)$$ $$L(1)$$ $$\approx$$ $$0.2313659624$$ $$L(\frac12)$$ $$\approx$$ $$0.2313659624$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4$with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
3$C_2$ $$1 + p T^{2}$$
7$C_2$ $$1 - T + T^{2}$$
good5$C_2^2$ $$1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4}$$
11$C_2^2$ $$1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
13$C_2^2$ $$1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4}$$
17$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
19$C_2$ $$( 1 + 7 T + p T^{2} )^{2}$$
23$C_2^2$ $$1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4}$$
29$C_2^2$ $$1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
31$C_2^2$ $$1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4}$$
37$C_2$ $$( 1 + 6 T + p T^{2} )^{2}$$
41$C_2^2$ $$1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
43$C_2^2$ $$1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4}$$
47$C_2^2$ $$1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4}$$
53$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
59$C_2^2$ $$1 - p T^{2} + p^{2} T^{4}$$
61$C_2^2$ $$1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4}$$
67$C_2^2$ $$1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4}$$
71$C_2$ $$( 1 + 15 T + p T^{2} )^{2}$$
73$C_2$ $$( 1 + 2 T + p T^{2} )^{2}$$
79$C_2^2$ $$1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4}$$
83$C_2^2$ $$1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4}$$
89$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
97$C_2^2$ $$1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4}$$
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\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}

## Imaginary part of the first few zeros on the critical line

−10.41334741536988331336256046221, −9.975751279554894950698185171397, −9.117398423675225066619430229700, −8.796483284033356513921534811987, −8.709501614789002665773507504302, −8.019502367706033541894703541010, −7.890511484569878029122806067196, −7.10000505504824821251188717567, −6.94412296329149120012633734727, −6.10039384850856176607967336767, −6.08688729004508515663104924858, −5.38471656937120029026824900334, −4.79050826078438600298430538030, −4.67022628736874351729717458984, −4.20506971988925797770657569798, −3.11584046593279726622395732545, −2.68656046954670565157106998634, −2.27866574135177906016483310826, −1.89066272747593138913215682996, −0.19802837887515089557502944622, 0.19802837887515089557502944622, 1.89066272747593138913215682996, 2.27866574135177906016483310826, 2.68656046954670565157106998634, 3.11584046593279726622395732545, 4.20506971988925797770657569798, 4.67022628736874351729717458984, 4.79050826078438600298430538030, 5.38471656937120029026824900334, 6.08688729004508515663104924858, 6.10039384850856176607967336767, 6.94412296329149120012633734727, 7.10000505504824821251188717567, 7.890511484569878029122806067196, 8.019502367706033541894703541010, 8.709501614789002665773507504302, 8.796483284033356513921534811987, 9.117398423675225066619430229700, 9.975751279554894950698185171397, 10.41334741536988331336256046221