Properties

Degree 2
Conductor $ 5 \cdot 6163 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 2·3-s + 2·4-s − 5-s + 4·6-s − 4·7-s + 9-s + 2·10-s − 4·11-s − 4·12-s − 6·13-s + 8·14-s + 2·15-s − 4·16-s − 5·17-s − 2·18-s − 8·19-s − 2·20-s + 8·21-s + 8·22-s − 23-s + 25-s + 12·26-s + 4·27-s − 8·28-s + 2·29-s − 4·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 4-s − 0.447·5-s + 1.63·6-s − 1.51·7-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 1.15·12-s − 1.66·13-s + 2.13·14-s + 0.516·15-s − 16-s − 1.21·17-s − 0.471·18-s − 1.83·19-s − 0.447·20-s + 1.74·21-s + 1.70·22-s − 0.208·23-s + 1/5·25-s + 2.35·26-s + 0.769·27-s − 1.51·28-s + 0.371·29-s − 0.730·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 30815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 30815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(30815\)    =    \(5 \cdot 6163\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{30815} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 30815,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;6163\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;6163\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 + T \)
6163 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.25967416764369, −15.58682423770373, −15.16283535357406, −14.57268231515949, −13.42544609609739, −13.15101167010216, −12.53295981754807, −12.17955905646106, −11.39997941159636, −10.92177162949891, −10.46261294382832, −10.03572729216724, −9.602433376285100, −8.881056320664509, −8.393094896673548, −7.716722493577355, −7.153407358742118, −6.527732355437050, −6.325444458204868, −5.390552286993094, −4.602926237710256, −4.335759867574972, −2.904436677078434, −2.608707027964231, −1.620195570995958, 0, 0, 0, 1.620195570995958, 2.608707027964231, 2.904436677078434, 4.335759867574972, 4.602926237710256, 5.390552286993094, 6.325444458204868, 6.527732355437050, 7.153407358742118, 7.716722493577355, 8.393094896673548, 8.881056320664509, 9.602433376285100, 10.03572729216724, 10.46261294382832, 10.92177162949891, 11.39997941159636, 12.17955905646106, 12.53295981754807, 13.15101167010216, 13.42544609609739, 14.57268231515949, 15.16283535357406, 15.58682423770373, 16.25967416764369

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