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Properties

Label 2-130-65.58-c1-0-1
Degree $2$
Conductor $130$
Sign $0.996 + 0.0880i$
Analytic cond. $1.03805$
Root an. cond. $1.01884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (−2.63 + 0.706i)3-s + (0.499 + 0.866i)4-s + (0.892 − 2.05i)5-s + (2.63 + 0.706i)6-s + (1.79 + 3.11i)7-s − 0.999i·8-s + (3.84 − 2.22i)9-s + (−1.79 + 1.32i)10-s + (4.64 − 1.24i)11-s + (−1.92 − 1.92i)12-s + (3.33 − 1.37i)13-s − 3.59i·14-s + (−0.904 + 6.03i)15-s + (−0.5 + 0.866i)16-s + (−0.819 + 3.05i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−1.52 + 0.407i)3-s + (0.249 + 0.433i)4-s + (0.399 − 0.916i)5-s + (1.07 + 0.288i)6-s + (0.679 + 1.17i)7-s − 0.353i·8-s + (1.28 − 0.740i)9-s + (−0.568 + 0.420i)10-s + (1.39 − 0.374i)11-s + (−0.556 − 0.556i)12-s + (0.924 − 0.381i)13-s − 0.961i·14-s + (−0.233 + 1.55i)15-s + (−0.125 + 0.216i)16-s + (−0.198 + 0.741i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.996 + 0.0880i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.996 + 0.0880i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629796 - 0.0277912i\)
\(L(\frac12)\) \(\approx\) \(0.629796 - 0.0277912i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.892 + 2.05i)T \)
13 \( 1 + (-3.33 + 1.37i)T \)
good3 \( 1 + (2.63 - 0.706i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.79 - 3.11i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.64 + 1.24i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.819 - 3.05i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.782 - 2.91i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.00 + 3.75i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-4.52 - 2.61i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.00 - 4.00i)T - 31iT^{2} \)
37 \( 1 + (-2.55 + 4.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.124 + 0.463i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.77 - 0.743i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 7.17T + 47T^{2} \)
53 \( 1 + (2.20 + 2.20i)T + 53iT^{2} \)
59 \( 1 + (10.0 + 2.68i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.826 - 1.43i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.02 + 1.74i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.32 + 2.49i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 + 1.21iT - 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + (2.64 + 9.88i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (9.00 - 5.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.62282366843637948050407988896, −12.13176177409076442223380687687, −11.25734721673312911505114909605, −10.40865493296274556318951527415, −9.064407599869041412229204665572, −8.449229200687544640544106901210, −6.29288434632604888657203958169, −5.62313640650976256290361393798, −4.24677085692666030153089641074, −1.40398146015036678220760443720, 1.31904551889943623662204075573, 4.36586956147131041646988158611, 5.98171593257860625965616593026, 6.77728289606802575242039958158, 7.46741378931546924884356521270, 9.324463324008642505727474202051, 10.50264929974863661657415206649, 11.23257078767463634213897279558, 11.76101968086056614218056564069, 13.50972909906339403516009023772

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