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 + 2·3-s + 4-s − 5-s + 2·6-s − 4·7-s + 8-s + 9-s − 10-s − 2·11-s + 2·12-s − 13-s − 4·14-s − 2·15-s + 16-s + 2·17-s + 18-s + 6·19-s − 20-s − 8·21-s − 2·22-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 − 0.603·11-s + 0.577·12-s − 0.277·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 1.74·21-s − 0.426·22-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  =  0
Selberg data  =  $(2,\ 130,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.773094903$
$L(\frac12)$  $\approx$  $1.773094903$
$L(\frac{3}{2})$   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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
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

−19.71673992637265, −19.37082981001345, −18.31346788068952, −16.63865123221418, −15.95328643335043, −15.11017032694798, −14.26094937907612, −13.26679558606258, −12.71202338625230, −11.47889106396016, −10.04470252993954, −9.161137327460865, −7.835317438248001, −6.875513157076544, −5.339768920502473, −3.530093107548732, −2.882539717059177, 2.882539717059177, 3.530093107548732, 5.339768920502473, 6.875513157076544, 7.835317438248001, 9.161137327460865, 10.04470252993954, 11.47889106396016, 12.71202338625230, 13.26679558606258, 14.26094937907612, 15.11017032694798, 15.95328643335043, 16.63865123221418, 18.31346788068952, 19.37082981001345, 19.71673992637265

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