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Properties

Label 2-130-65.28-c1-0-2
Degree $2$
Conductor $130$
Sign $0.950 + 0.310i$
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 + (−0.0387 + 0.144i)3-s + (0.499 − 0.866i)4-s + (0.105 − 2.23i)5-s + (−0.0387 − 0.144i)6-s + (1.10 − 1.90i)7-s + 0.999i·8-s + (2.57 + 1.48i)9-s + (1.02 + 1.98i)10-s + (0.775 − 2.89i)11-s + (0.105 + 0.105i)12-s + (−0.759 + 3.52i)13-s + 2.20i·14-s + (0.319 + 0.101i)15-s + (−0.5 − 0.866i)16-s + (−1.13 + 0.303i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.0223 + 0.0835i)3-s + (0.249 − 0.433i)4-s + (0.0473 − 0.998i)5-s + (−0.0158 − 0.0590i)6-s + (0.416 − 0.721i)7-s + 0.353i·8-s + (0.859 + 0.496i)9-s + (0.324 + 0.628i)10-s + (0.233 − 0.872i)11-s + (0.0305 + 0.0305i)12-s + (−0.210 + 0.977i)13-s + 0.589i·14-s + (0.0823 + 0.0263i)15-s + (−0.125 − 0.216i)16-s + (−0.274 + 0.0736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.950 + 0.310i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.950 + 0.310i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.868037 - 0.138092i\)
\(L(\frac12)\) \(\approx\) \(0.868037 - 0.138092i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.105 + 2.23i)T \)
13 \( 1 + (0.759 - 3.52i)T \)
good3 \( 1 + (0.0387 - 0.144i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.10 + 1.90i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.775 + 2.89i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.13 - 0.303i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-7.28 + 1.95i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (6.12 + 1.64i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.05 - 1.76i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.08 - 5.08i)T - 31iT^{2} \)
37 \( 1 + (-4.19 - 7.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.89 + 0.775i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.11 + 4.16i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 3.09T + 47T^{2} \)
53 \( 1 + (-9.43 - 9.43i)T + 53iT^{2} \)
59 \( 1 + (-0.500 - 1.86i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.02 - 1.77i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.33 + 4.23i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.665 + 2.48i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 4.39iT - 73T^{2} \)
79 \( 1 - 13.8iT - 79T^{2} \)
83 \( 1 + 4.99T + 83T^{2} \)
89 \( 1 + (1.78 + 0.478i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.83 + 4.52i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.55265400424951616729893761419, −12.10965918525604097975353415691, −11.11274655361064103399646837304, −9.953192943886167911370388801119, −9.040648601831200240179406820355, −7.956444804928781337113320775291, −6.97041342531876182741908208591, −5.37613182255604524160888247055, −4.19853590948820991476716379269, −1.41512676808580650492133957501, 2.09784444105126655131885217082, 3.72092262287735982649051841937, 5.69793717836458202342290213091, 7.15071171325213456282288462137, 7.88514845887763469296419595477, 9.575531936452724288419703404440, 10.01230713730792614703096485142, 11.37072644754716017071039111001, 12.08721798417875952493438521060, 13.15282596008724507601880889689

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