Properties

Degree 2
Conductor $ 2 \cdot 5 \cdot 13 $
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 + 4-s + 5-s + 8-s − 3·9-s + 10-s + 13-s + 16-s + 2·17-s − 3·18-s − 8·19-s + 20-s − 4·23-s + 25-s + 26-s − 2·29-s − 4·31-s + 32-s + 2·34-s − 3·36-s + 6·37-s − 8·38-s + 40-s + 10·41-s − 3·45-s − 4·46-s + 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s − 9-s + 0.316·10-s + 0.277·13-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 1.83·19-s + 0.223·20-s − 0.834·23-s + 1/5·25-s + 0.196·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 0.986·37-s − 1.29·38-s + 0.158·40-s + 1.56·41-s − 0.447·45-s − 0.589·46-s + 1.16·47-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  =  0
Selberg data  =  $(2,\ 130,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.564713754$
$L(\frac12)$  $\approx$  $1.564713754$
$L(\frac{3}{2})$   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 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−19.93238314244035, −19.06135467102217, −17.86813092150381, −16.97396693442125, −16.20153396054195, −14.88181490712605, −14.36479745251109, −13.31482683467550, −12.46224067291033, −11.33556569842126, −10.46707546118065, −9.084106232831127, −7.923578663951582, −6.375975254259472, −5.569059877342025, −4.026390038085724, −2.402921815149482, 2.402921815149482, 4.026390038085724, 5.569059877342025, 6.375975254259472, 7.923578663951582, 9.084106232831127, 10.46707546118065, 11.33556569842126, 12.46224067291033, 13.31482683467550, 14.36479745251109, 14.88181490712605, 16.20153396054195, 16.97396693442125, 17.86813092150381, 19.06135467102217, 19.93238314244035

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