Properties

Label 2-130-1.1-c1-0-2
Degree $2$
Conductor $130$
Sign $1$
Analytic cond. $1.03805$
Root an. cond. $1.01884$
Motivic weight $1$
Arithmetic yes
Rational yes
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

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{130} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.773094903\)
\(L(\frac12)\) \(\approx\) \(1.773094903\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.26679558606257756188931391942, −12.71202338625229541571828648444, −11.47889106396016028805556897933, −10.04470252993954468916012522294, −9.161137327460865013114506016469, −7.83531743824800124245932651769, −6.87551315707654393130092819540, −5.33976892050247260934145252130, −3.53009310754873249134401547682, −2.88253971705917726242541361642, 2.88253971705917726242541361642, 3.53009310754873249134401547682, 5.33976892050247260934145252130, 6.87551315707654393130092819540, 7.83531743824800124245932651769, 9.161137327460865013114506016469, 10.04470252993954468916012522294, 11.47889106396016028805556897933, 12.71202338625229541571828648444, 13.26679558606257756188931391942

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