Properties

Degree $4$
Conductor $394$
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-s − 2·3-s + 4-s − 4·5-s + 2·6-s + 7-s + 8-s + 2·9-s + 4·10-s + 2·11-s − 2·12-s − 2·13-s − 14-s + 8·15-s − 3·16-s − 2·17-s − 2·18-s + 19-s − 4·20-s − 2·21-s − 2·22-s + 5·23-s − 2·24-s + 6·25-s + 2·26-s − 6·27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 2/3·9-s + 1.26·10-s + 0.603·11-s − 0.577·12-s − 0.554·13-s − 0.267·14-s + 2.06·15-s − 3/4·16-s − 0.485·17-s − 0.471·18-s + 0.229·19-s − 0.894·20-s − 0.436·21-s − 0.426·22-s + 1.04·23-s − 0.408·24-s + 6/5·25-s + 0.392·26-s − 1.15·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 394 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 394 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(394\)    =    \(2 \cdot 197\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{394} (1, \cdot )$
Sato-Tate group: $\mathrm{USp}(4)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 394,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2007827436\)
\(L(\frac12)\) \(\approx\) \(0.2007827436\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + p T + p T^{2} ) \)
197$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 18 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
7$D_{4}$ \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 3 T + 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - T - 6 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
61$D_{4}$ \( 1 + T + 16 T^{2} + p T^{3} + p^{2} T^{4} \)
67$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$D_{4}$ \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T - 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 9 T + 66 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.6295715505, −19.3059469824, −19.0248356528, −17.9947296559, −17.5176256677, −16.8487865706, −16.5231674940, −15.6004576954, −15.4443043074, −14.5647329690, −13.5078710086, −12.5489303651, −11.7483015383, −11.4806467239, −10.9979364628, −10.0972223242, −9.01903323173, −8.03781843767, −7.36037006983, −6.65135433456, −5.09410049950, −4.06198103368, 4.06198103368, 5.09410049950, 6.65135433456, 7.36037006983, 8.03781843767, 9.01903323173, 10.0972223242, 10.9979364628, 11.4806467239, 11.7483015383, 12.5489303651, 13.5078710086, 14.5647329690, 15.4443043074, 15.6004576954, 16.5231674940, 16.8487865706, 17.5176256677, 17.9947296559, 19.0248356528, 19.3059469824, 19.6295715505

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