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Properties

Label 2-130-65.29-c1-0-2
Degree $2$
Conductor $130$
Sign $0.969 + 0.244i$
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 + (1.91 − 1.10i)3-s + (0.499 − 0.866i)4-s + (2.21 + 0.311i)5-s + (−1.10 + 1.91i)6-s + (−3.38 − 1.95i)7-s + 0.999i·8-s + (0.951 − 1.64i)9-s + (−2.07 + 0.837i)10-s + (−0.533 − 0.923i)11-s − 2.21i·12-s + (2.24 + 2.82i)13-s + 3.90·14-s + (4.59 − 1.85i)15-s + (−0.5 − 0.866i)16-s + (1.13 + 0.655i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (1.10 − 0.639i)3-s + (0.249 − 0.433i)4-s + (0.990 + 0.139i)5-s + (−0.451 + 0.782i)6-s + (−1.27 − 0.737i)7-s + 0.353i·8-s + (0.317 − 0.549i)9-s + (−0.655 + 0.264i)10-s + (−0.160 − 0.278i)11-s − 0.639i·12-s + (0.622 + 0.782i)13-s + 1.04·14-s + (1.18 − 0.478i)15-s + (−0.125 − 0.216i)16-s + (0.275 + 0.158i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.969 + 0.244i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.969 + 0.244i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14278 - 0.141568i\)
\(L(\frac12)\) \(\approx\) \(1.14278 - 0.141568i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 - 0.311i)T \)
13 \( 1 + (-2.24 - 2.82i)T \)
good3 \( 1 + (-1.91 + 1.10i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (3.38 + 1.95i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.533 + 0.923i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.13 - 0.655i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.29 - 5.70i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.85 + 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.52 + 7.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 + (4.34 - 2.51i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.97 - 5.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.73 + i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.0967iT - 47T^{2} \)
53 \( 1 + 1.49iT - 53T^{2} \)
59 \( 1 + (-3.59 + 6.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.94 - 6.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.29 + 4.21i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.73 + 13.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 15.3iT - 73T^{2} \)
79 \( 1 + 4.30T + 79T^{2} \)
83 \( 1 - 9.69iT - 83T^{2} \)
89 \( 1 + (-2.26 - 3.91i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.68 + 2.13i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.47210066711066881338236441181, −12.71435075707999069328683570771, −10.83880650342951031488789877075, −9.833801192388689291220912159563, −9.075648922117588171691506179329, −7.976583690426684582839854768920, −6.82808030189361353884971883909, −6.00765668447740868350728261397, −3.50100618358977094855522680672, −1.90041423847207133885612956962, 2.46441569120002828839731081661, 3.45095706748229326479479550843, 5.54100489196453246920650755900, 6.96409850927819276835921735757, 8.725761958214022609796765517381, 9.117292137157584069585988137965, 9.925541008911276581153600929720, 10.88121462076270881214777272776, 12.71195500854466702251089350278, 13.09823118744110600383328719453

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