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Properties

Label 2-130-65.23-c2-0-10
Degree $2$
Conductor $130$
Sign $-0.365 + 0.930i$
Analytic cond. $3.54224$
Root an. cond. $1.88208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 1.36i)2-s + (0.957 − 3.57i)3-s + (−1.73 − i)4-s + (4.99 + 0.300i)5-s + (−4.52 − 2.61i)6-s + (9.12 − 2.44i)7-s + (−2 + 1.99i)8-s + (−4.04 − 2.33i)9-s + (2.23 − 6.70i)10-s + (−15.1 + 8.74i)11-s + (−5.22 + 5.22i)12-s + (−6.39 − 11.3i)13-s − 13.3i·14-s + (5.85 − 17.5i)15-s + (1.99 + 3.46i)16-s + (1.68 + 6.27i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.319 − 1.19i)3-s + (−0.433 − 0.250i)4-s + (0.998 + 0.0601i)5-s + (−0.754 − 0.435i)6-s + (1.30 − 0.349i)7-s + (−0.250 + 0.249i)8-s + (−0.449 − 0.259i)9-s + (0.223 − 0.670i)10-s + (−1.37 + 0.795i)11-s + (−0.435 + 0.435i)12-s + (−0.491 − 0.870i)13-s − 0.954i·14-s + (0.390 − 1.16i)15-s + (0.124 + 0.216i)16-s + (0.0989 + 0.369i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $-0.365 + 0.930i$
Analytic conductor: \(3.54224\)
Root analytic conductor: \(1.88208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1),\ -0.365 + 0.930i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.08362 - 1.58952i\)
\(L(\frac12)\) \(\approx\) \(1.08362 - 1.58952i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
5 \( 1 + (-4.99 - 0.300i)T \)
13 \( 1 + (6.39 + 11.3i)T \)
good3 \( 1 + (-0.957 + 3.57i)T + (-7.79 - 4.5i)T^{2} \)
7 \( 1 + (-9.12 + 2.44i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (15.1 - 8.74i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + (-1.68 - 6.27i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (16.0 - 27.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (6.59 - 24.6i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + (-43.3 + 25.0i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 31.1iT - 961T^{2} \)
37 \( 1 + (2.27 - 8.50i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-3.28 + 1.89i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (40.7 - 10.9i)T + (1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-34.7 - 34.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (26.8 + 26.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (32.8 - 56.9i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (22.4 - 38.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-11.6 + 43.5i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-29.4 - 17.0i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-6.68 + 6.68i)T - 5.32e3iT^{2} \)
79 \( 1 + 137. iT - 6.24e3T^{2} \)
83 \( 1 + (29.6 - 29.6i)T - 6.88e3iT^{2} \)
89 \( 1 + (-5.93 - 10.2i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-40.0 + 10.7i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−12.85532647786418071963437869670, −12.03615094098272617770613564802, −10.55284294773478768114389518912, −10.01104687295163898920435694832, −8.154860056643869614462042125774, −7.66964776963599867621572035323, −5.95538196439154870560878454972, −4.74950477434434299990862090177, −2.45892792081836345899663808527, −1.55175565812472029907987918625, 2.63938350249961095528946364449, 4.76543363410302259569638432893, 5.09096416219998150132989338929, 6.72232435085316746055729602122, 8.417547799960797786280670864059, 8.959092978850381493321391997043, 10.23918059295592119441473056160, 11.00333580514984300608151456110, 12.58012714110165498320037278118, 13.88541141212203776973256981949

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