Properties

Degree 4
Conductor $ 2^{3} \cdot 1907 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 2

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4·3-s + 4-s − 4·5-s + 4·6-s − 3·7-s − 8-s + 7·9-s + 4·10-s − 4·11-s − 4·12-s − 4·13-s + 3·14-s + 16·15-s + 16-s − 4·17-s − 7·18-s − 4·20-s + 12·21-s + 4·22-s − 6·23-s + 4·24-s + 7·25-s + 4·26-s − 4·27-s − 3·28-s + 3·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 2.30·3-s + 1/2·4-s − 1.78·5-s + 1.63·6-s − 1.13·7-s − 0.353·8-s + 7/3·9-s + 1.26·10-s − 1.20·11-s − 1.15·12-s − 1.10·13-s + 0.801·14-s + 4.13·15-s + 1/4·16-s − 0.970·17-s − 1.64·18-s − 0.894·20-s + 2.61·21-s + 0.852·22-s − 1.25·23-s + 0.816·24-s + 7/5·25-s + 0.784·26-s − 0.769·27-s − 0.566·28-s + 0.557·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 15256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n\]
\[\begin{aligned} \Lambda(s)=\mathstrut & 15256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(15256\)    =    \(2^{3} \cdot 1907\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{15256} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 15256,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;1907\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{2,\;1907\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( 1 + T \)
1907$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 29 T + p T^{2} ) \)
good3$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$D_{4}$ \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + 6 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 6 T + 29 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + 35 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 52 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 5 T + 74 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T - 65 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 52 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 9 T + 123 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T - 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 23 T + 3 p T^{2} - 23 p T^{3} + p^{2} T^{4} \)
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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

−16.6663770728, −16.0156219306, −15.8708779654, −15.5733748636, −15.0242315898, −14.2480261738, −13.343611601, −12.7864692444, −12.2670515695, −12.0717864966, −11.6209805502, −11.0905606106, −10.7820988842, −10.176011687, −9.81105534598, −8.92909993499, −8.13583138126, −7.6652001916, −7.11424039809, −6.44070593676, −6.09208051893, −5.17137598374, −4.72042419532, −3.79323740141, −2.6980436659, 0, 0, 2.6980436659, 3.79323740141, 4.72042419532, 5.17137598374, 6.09208051893, 6.44070593676, 7.11424039809, 7.6652001916, 8.13583138126, 8.92909993499, 9.81105534598, 10.176011687, 10.7820988842, 11.0905606106, 11.6209805502, 12.0717864966, 12.2670515695, 12.7864692444, 13.343611601, 14.2480261738, 15.0242315898, 15.5733748636, 15.8708779654, 16.0156219306, 16.6663770728

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