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Properties

Label 2-130-65.19-c2-0-4
Degree $2$
Conductor $130$
Sign $0.978 + 0.206i$
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  + (1.36 − 0.366i)2-s + (−1.64 − 0.951i)3-s + (1.73 − i)4-s + (1.64 + 4.72i)5-s + (−2.60 − 0.696i)6-s + (11.1 + 2.98i)7-s + (1.99 − 2i)8-s + (−2.68 − 4.65i)9-s + (3.97 + 5.84i)10-s + (4.34 − 1.16i)11-s − 3.80·12-s + (3.12 − 12.6i)13-s + 16.3·14-s + (1.78 − 9.35i)15-s + (1.99 − 3.46i)16-s + (8.80 + 15.2i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.549 − 0.317i)3-s + (0.433 − 0.250i)4-s + (0.328 + 0.944i)5-s + (−0.433 − 0.116i)6-s + (1.59 + 0.427i)7-s + (0.249 − 0.250i)8-s + (−0.298 − 0.517i)9-s + (0.397 + 0.584i)10-s + (0.395 − 0.105i)11-s − 0.317·12-s + (0.240 − 0.970i)13-s + 1.16·14-s + (0.118 − 0.623i)15-s + (0.124 − 0.216i)16-s + (0.518 + 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.978 + 0.206i$
Analytic conductor: \(3.54224\)
Root analytic conductor: \(1.88208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1),\ 0.978 + 0.206i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.02119 - 0.211287i\)
\(L(\frac12)\) \(\approx\) \(2.02119 - 0.211287i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
5 \( 1 + (-1.64 - 4.72i)T \)
13 \( 1 + (-3.12 + 12.6i)T \)
good3 \( 1 + (1.64 + 0.951i)T + (4.5 + 7.79i)T^{2} \)
7 \( 1 + (-11.1 - 2.98i)T + (42.4 + 24.5i)T^{2} \)
11 \( 1 + (-4.34 + 1.16i)T + (104. - 60.5i)T^{2} \)
17 \( 1 + (-8.80 - 15.2i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (17.6 + 4.73i)T + (312. + 180.5i)T^{2} \)
23 \( 1 + (7.54 - 13.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (3.26 - 5.65i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (20.0 - 20.0i)T - 961iT^{2} \)
37 \( 1 + (12.7 + 47.5i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (15.3 + 57.3i)T + (-1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (8.81 + 15.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (30.5 - 30.5i)T - 2.20e3iT^{2} \)
53 \( 1 + 49.7iT - 2.80e3T^{2} \)
59 \( 1 + (9.71 - 36.2i)T + (-3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (22.2 + 38.5i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-32.1 + 8.62i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (61.4 + 16.4i)T + (4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (67.5 - 67.5i)T - 5.32e3iT^{2} \)
79 \( 1 + 139.T + 6.24e3T^{2} \)
83 \( 1 + (-80.8 - 80.8i)T + 6.88e3iT^{2} \)
89 \( 1 + (-152. + 40.9i)T + (6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (1.66 - 6.20i)T + (-8.14e3 - 4.70e3i)T^{2} \)
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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

−12.90601302016714919710139711249, −11.95133984178650931480979230595, −11.12087186279663650910458856221, −10.47940375457546227085994790644, −8.702259897761502931021493788280, −7.39856027997088782724600896867, −6.10019176660507098766982200002, −5.36277714569037194911716335624, −3.58452190724883834924470238324, −1.81617462570389623697164111176, 1.76221013004627018460180098317, 4.43059603104870429283507447129, 4.89177513730139606344250174039, 6.13949510217359293461266309853, 7.74619726815485934787456973796, 8.700634482070358037747401928322, 10.22025435552083645401344819997, 11.45670608878138901024150906935, 11.82768461396583940966650619611, 13.27481150904931165881214240419

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