Properties

Degree 4
Conductor 353
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 − 2·4-s + 5-s + 2·6-s + 3·8-s − 10-s + 2·11-s + 4·12-s − 13-s − 2·15-s + 16-s + 6·17-s − 6·19-s − 2·20-s − 2·22-s − 2·23-s − 6·24-s − 25-s + 26-s + 2·27-s − 2·29-s + 2·30-s − 4·31-s − 2·32-s − 4·33-s − 6·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s − 4-s + 0.447·5-s + 0.816·6-s + 1.06·8-s − 0.316·10-s + 0.603·11-s + 1.15·12-s − 0.277·13-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 1.37·19-s − 0.447·20-s − 0.426·22-s − 0.417·23-s − 1.22·24-s − 1/5·25-s + 0.196·26-s + 0.384·27-s − 0.371·29-s + 0.365·30-s − 0.718·31-s − 0.353·32-s − 0.696·33-s − 1.02·34-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 353 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 353 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(353\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{353} (1, \cdot )$
Sato-Tate  :  $\mathrm{USp}(4)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(4,\ 353,\ (\ :1/2, 1/2),\ 1)$
$L(1)$  $\approx$  $0.1859131786$
$L(\frac12)$  $\approx$  $0.1859131786$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 353$, \[F_p(T) = 1 - a_p T + b_p T^2 - a_p p T^3 + p^2 T^4 \]with $b_p = a_p^2 - a_{p^2}$. If $p = 353$, then $F_p$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p$
bad353$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 9 T + p T^{2} ) \)
good2$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
3$D_{4}$ \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
5$D_{4}$ \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \)
7$V_4$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \)
19$D_{4}$ \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
31$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
41$D_{4}$ \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 75 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$V_4$ \( 1 + 23 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 12 T + 142 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$V_4$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 6 T + 92 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 10 T + 74 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 14 T + 140 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
83$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
89$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 11 T + 116 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
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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

−19.9961442661, −19.3214919249, −18.7976671865, −18.209207474, −17.6758825494, −17.114144005, −16.8700603189, −16.3062469438, −15.0721190988, −14.3933411178, −13.8616163752, −12.9097577259, −12.352638662, −11.505940061, −10.7395302821, −9.9438356318, −9.33983492188, −8.54105304961, −7.57074114369, −6.1675392457, −5.48946407918, −4.23852001252, 4.23852001252, 5.48946407918, 6.1675392457, 7.57074114369, 8.54105304961, 9.33983492188, 9.9438356318, 10.7395302821, 11.505940061, 12.352638662, 12.9097577259, 13.8616163752, 14.3933411178, 15.0721190988, 16.3062469438, 16.8700603189, 17.114144005, 17.6758825494, 18.209207474, 18.7976671865, 19.3214919249, 19.9961442661

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