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Properties

Label 2-130-65.9-c1-0-0
Degree $2$
Conductor $130$
Sign $-0.491 - 0.870i$
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 + (−1.45 − 0.837i)3-s + (0.499 + 0.866i)4-s + (−1.67 + 1.48i)5-s + (0.837 + 1.45i)6-s + (−1.56 + 0.903i)7-s − 0.999i·8-s + (−0.0969 − 0.167i)9-s + (2.19 − 0.445i)10-s + (−3.22 + 5.58i)11-s − 1.67i·12-s + (1.98 + 3.01i)13-s + 1.80·14-s + (3.67 − 0.745i)15-s + (−0.5 + 0.866i)16-s + (−0.416 + 0.240i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.837 − 0.483i)3-s + (0.249 + 0.433i)4-s + (−0.749 + 0.662i)5-s + (0.341 + 0.592i)6-s + (−0.591 + 0.341i)7-s − 0.353i·8-s + (−0.0323 − 0.0559i)9-s + (0.692 − 0.140i)10-s + (−0.971 + 1.68i)11-s − 0.483i·12-s + (0.549 + 0.835i)13-s + 0.482·14-s + (0.947 − 0.192i)15-s + (−0.125 + 0.216i)16-s + (−0.101 + 0.0583i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 - 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $-0.491 - 0.870i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ -0.491 - 0.870i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0972584 + 0.166575i\)
\(L(\frac12)\) \(\approx\) \(0.0972584 + 0.166575i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 1.48i)T \)
13 \( 1 + (-1.98 - 3.01i)T \)
good3 \( 1 + (1.45 + 0.837i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.56 - 0.903i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.22 - 5.58i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.416 - 0.240i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.14 + 5.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.16 + 3.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.15 + 2.00i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.25T + 31T^{2} \)
37 \( 1 + (-2.65 - 1.53i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.75 - 6.50i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.73 - i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 2.19iT - 47T^{2} \)
53 \( 1 + 0.906iT - 53T^{2} \)
59 \( 1 + (-3.28 - 5.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.47 - 9.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.562 - 0.324i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.83 + 3.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.60iT - 73T^{2} \)
79 \( 1 - 2.29T + 79T^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 + (0.578 - 1.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.9 - 6.91i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.21649971949571607167043885212, −12.28966583420773465409084775563, −11.66998745509272851418069371828, −10.67328718118855702779522137749, −9.694697668445695917716154903334, −8.297199005909610440721656790085, −7.02890718985314063814149522073, −6.38940122940346003693385880164, −4.40342043906863960155384983894, −2.51101376793168525427512819477, 0.25574969252288303677080229989, 3.64269483012785613488406966416, 5.34889177952392950443688157828, 6.11041424838698290980474455987, 7.946241813359188703932049555735, 8.448509888422641274838908377615, 10.08489574622713325633360135280, 10.77673907067060345720453112582, 11.66039132946720135584574536401, 12.91196082567534504238493206812

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