Properties

Degree 4
Conductor $ 149 \cdot 673 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·3-s + 2·4-s + 2·5-s + 4·6-s − 2·7-s + 4·8-s + 2·9-s + 4·10-s − 5·11-s + 4·12-s − 3·13-s − 4·14-s + 4·15-s + 8·16-s − 2·17-s + 4·18-s + 4·19-s + 4·20-s − 4·21-s − 10·22-s + 5·23-s + 8·24-s + 2·25-s − 6·26-s + 6·27-s − 4·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 4-s + 0.894·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 2/3·9-s + 1.26·10-s − 1.50·11-s + 1.15·12-s − 0.832·13-s − 1.06·14-s + 1.03·15-s + 2·16-s − 0.485·17-s + 0.942·18-s + 0.917·19-s + 0.894·20-s − 0.872·21-s − 2.13·22-s + 1.04·23-s + 1.63·24-s + 2/5·25-s − 1.17·26-s + 1.15·27-s − 0.755·28-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  =  0
Selberg data  =  $(4,\ 100277,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $5.063458818$
$L(\frac12)$  $\approx$  $5.063458818$
$L(\frac{3}{2})$   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$ \( ( 1 + T )( 1 - 9 T + p T^{2} ) \)
673$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 41 T + p T^{2} ) \)
good2$V_4$ \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \)
3$V_4$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$V_4$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 5 T + 19 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$D_{4}$ \( 1 + 2 T - 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 5 T + 24 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 5 T + 9 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + T + 56 T^{2} + p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 3 T - 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 5 T + 68 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - T + 80 T^{2} - p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 59 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 4 T - 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 17 T + 170 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - T + 58 T^{2} - p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 10 T + 160 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
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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

−13.9474051464, −13.5312582377, −13.3350792872, −13.0167716432, −12.5933001606, −12.2380024832, −11.4964198713, −10.8822128938, −10.3985108225, −10.1400538147, −9.50339092291, −9.32205457854, −8.39352811093, −8.11732528511, −7.43992627883, −7.03057676304, −6.5849057483, −5.64718744355, −5.39225260525, −4.76566931995, −4.37622840695, −3.42544924403, −2.80656417728, −2.65398446953, −1.5774917723, 1.5774917723, 2.65398446953, 2.80656417728, 3.42544924403, 4.37622840695, 4.76566931995, 5.39225260525, 5.64718744355, 6.5849057483, 7.03057676304, 7.43992627883, 8.11732528511, 8.39352811093, 9.32205457854, 9.50339092291, 10.1400538147, 10.3985108225, 10.8822128938, 11.4964198713, 12.2380024832, 12.5933001606, 13.0167716432, 13.3350792872, 13.5312582377, 13.9474051464

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