Properties

Degree 4
Conductor $ 149 \cdot 673 $
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 − 2·3-s − 2·4-s − 2·5-s + 2·6-s − 3·7-s + 3·8-s + 2·9-s + 2·10-s + 11-s + 4·12-s − 13-s + 3·14-s + 4·15-s + 16-s − 2·17-s − 2·18-s + 4·20-s + 6·21-s − 22-s + 2·23-s − 6·24-s − 5·25-s + 26-s − 6·27-s + 6·28-s − 3·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 4-s − 0.894·5-s + 0.816·6-s − 1.13·7-s + 1.06·8-s + 2/3·9-s + 0.632·10-s + 0.301·11-s + 1.15·12-s − 0.277·13-s + 0.801·14-s + 1.03·15-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.894·20-s + 1.30·21-s − 0.213·22-s + 0.417·23-s − 1.22·24-s − 25-s + 0.196·26-s − 1.15·27-s + 1.13·28-s − 0.557·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 100277 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 100277 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(100277\)    =    \(149 \cdot 673\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{100277} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  2
Selberg data  =  $(4,\ 100277,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{149,\;673\}$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p \in \{149,\;673\}$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad149$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 6 T + p T^{2} ) \)
673$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 44 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
3$V_4$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$C_4$ \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$V_4$ \( 1 + 12 T^{2} + p^{2} T^{4} \)
23$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \)
29$D_{4}$ \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 9 T + 51 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
41$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$D_{4}$ \( 1 + 6 T + 75 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T - 89 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 10 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$V_4$ \( 1 + 52 T^{2} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
71$D_{4}$ \( 1 + 8 T + 146 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 10 T + 35 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 116 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 5 T - 20 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 11 T + 150 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
97$V_4$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
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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

−14.5313283919, −13.8174585929, −13.5771270725, −13.1172928081, −12.6596356057, −12.2970673201, −11.8134139345, −11.2999508887, −11.1028266278, −10.3378325837, −10.0133509138, −9.53067473702, −9.16406414732, −8.76281687216, −8.04679040661, −7.71037425608, −6.98194853414, −6.70426036273, −5.94233602139, −5.44846515949, −4.91683594441, −4.07881694741, −3.91157222741, −3.04724986928, −1.6254014106, 0, 0, 1.6254014106, 3.04724986928, 3.91157222741, 4.07881694741, 4.91683594441, 5.44846515949, 5.94233602139, 6.70426036273, 6.98194853414, 7.71037425608, 8.04679040661, 8.76281687216, 9.16406414732, 9.53067473702, 10.0133509138, 10.3378325837, 11.1028266278, 11.2999508887, 11.8134139345, 12.2970673201, 12.6596356057, 13.1172928081, 13.5771270725, 13.8174585929, 14.5313283919

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