Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 5-s + 2·6-s − 4·7-s − 8-s + 9-s − 10-s − 6·11-s − 2·12-s + 13-s + 4·14-s − 2·15-s + 16-s − 6·17-s − 18-s + 2·19-s + 20-s + 8·21-s + 6·22-s + 6·23-s + 2·24-s + 25-s − 26-s + 4·27-s − 4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.447·5-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s − 0.577·12-s + 0.277·13-s + 1.06·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s + 0.223·20-s + 1.74·21-s + 1.27·22-s + 1.25·23-s + 0.408·24-s + 1/5·25-s − 0.196·26-s + 0.769·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(130\)    =    \(2 \cdot 5 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{130} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 130,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−19.07640579694152, −18.27167125715934, −17.63206125623943, −16.62162295541461, −16.05347128900377, −15.25323276754618, −13.22444798801553, −12.96259457984179, −11.49466654996308, −10.65124822704304, −9.884499091813617, −8.747160169511579, −7.132840466877257, −6.218654120343354, −5.209120259795546, −2.835820446784278, 0, 2.835820446784278, 5.209120259795546, 6.218654120343354, 7.132840466877257, 8.747160169511579, 9.884499091813617, 10.65124822704304, 11.49466654996308, 12.96259457984179, 13.22444798801553, 15.25323276754618, 16.05347128900377, 16.62162295541461, 17.63206125623943, 18.27167125715934, 19.07640579694152

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