Properties

Degree 4
Conductor $ 5^{2} \cdot 4013 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4-s + 3·7-s − 13-s − 6·14-s + 16-s − 8·17-s − 7·19-s + 25-s + 2·26-s + 3·28-s + 6·29-s + 7·31-s + 2·32-s + 16·34-s + 6·37-s + 14·38-s − 3·41-s − 43-s − 47-s + 2·49-s − 2·50-s − 52-s − 2·53-s − 12·58-s − 5·59-s − 7·61-s + ⋯
L(s)  = 1  − 1.41·2-s + 1/2·4-s + 1.13·7-s − 0.277·13-s − 1.60·14-s + 1/4·16-s − 1.94·17-s − 1.60·19-s + 1/5·25-s + 0.392·26-s + 0.566·28-s + 1.11·29-s + 1.25·31-s + 0.353·32-s + 2.74·34-s + 0.986·37-s + 2.27·38-s − 0.468·41-s − 0.152·43-s − 0.145·47-s + 2/7·49-s − 0.282·50-s − 0.138·52-s − 0.274·53-s − 1.57·58-s − 0.650·59-s − 0.896·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100325 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100325 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100325\)    =    \(5^{2} \cdot 4013\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{100325} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(4,\ 100325,\ (\ :1/2, 1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;4013\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{5,\;4013\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
4013$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 83 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
3$V_4$ \( 1 + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \)
11$V_4$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 7 T + 34 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
23$V_4$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 32 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 7 T + 39 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 3 T + 19 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 5 T + 103 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 21 T + 245 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 5 T + 3 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 14 T + 172 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$D_{4}$ \( 1 - 6 T + 10 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−14.4843222278, −13.66969577, −13.507951761, −13.0217340345, −12.3477030036, −11.9953431733, −11.4873044049, −10.9621390683, −10.707369561, −10.230285197, −9.7090842957, −9.20257312162, −8.74568782093, −8.41036856876, −8.16532575253, −7.60459843109, −6.84964257506, −6.42955876414, −5.98322017496, −4.90248130383, −4.55576588026, −4.20290195031, −2.87411233258, −2.26428067794, −1.336450789, 0, 1.336450789, 2.26428067794, 2.87411233258, 4.20290195031, 4.55576588026, 4.90248130383, 5.98322017496, 6.42955876414, 6.84964257506, 7.60459843109, 8.16532575253, 8.41036856876, 8.74568782093, 9.20257312162, 9.7090842957, 10.230285197, 10.707369561, 10.9621390683, 11.4873044049, 11.9953431733, 12.3477030036, 13.0217340345, 13.507951761, 13.66969577, 14.4843222278

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