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Properties

Label 2-130-65.49-c1-0-5
Degree $2$
Conductor $130$
Sign $0.584 + 0.811i$
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  + (−0.5 − 0.866i)2-s + (2.08 − 1.20i)3-s + (−0.499 + 0.866i)4-s + (2.10 + 0.750i)5-s + (−2.08 − 1.20i)6-s + (−0.702 + 1.21i)7-s + 0.999·8-s + (1.40 − 2.43i)9-s + (−0.403 − 2.19i)10-s + (−4.59 + 2.65i)11-s + 2.40i·12-s + (−1.58 − 3.23i)13-s + 1.40·14-s + (5.30 − 0.972i)15-s + (−0.5 − 0.866i)16-s + (−6.09 − 3.52i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (1.20 − 0.695i)3-s + (−0.249 + 0.433i)4-s + (0.942 + 0.335i)5-s + (−0.852 − 0.491i)6-s + (−0.265 + 0.460i)7-s + 0.353·8-s + (0.467 − 0.810i)9-s + (−0.127 − 0.695i)10-s + (−1.38 + 0.800i)11-s + 0.695i·12-s + (−0.440 − 0.897i)13-s + 0.375·14-s + (1.36 − 0.251i)15-s + (−0.125 − 0.216i)16-s + (−1.47 − 0.853i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.584 + 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14884 - 0.588131i\)
\(L(\frac12)\) \(\approx\) \(1.14884 - 0.588131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-2.10 - 0.750i)T \)
13 \( 1 + (1.58 + 3.23i)T \)
good3 \( 1 + (-2.08 + 1.20i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.702 - 1.21i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.59 - 2.65i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (6.09 + 3.52i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.28 - 1.32i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.30 + 1.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.10 + 3.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.07iT - 31T^{2} \)
37 \( 1 + (-0.412 - 0.715i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.43 + 1.40i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.57 - 2.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.79T + 47T^{2} \)
53 \( 1 + 7.93iT - 53T^{2} \)
59 \( 1 + (-5.59 - 3.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.36 + 12.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.34 + 4.81i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.04T + 73T^{2} \)
79 \( 1 - 4.05T + 79T^{2} \)
83 \( 1 + 8.55T + 83T^{2} \)
89 \( 1 + (13.1 - 7.57i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.18 - 3.78i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.09927155617688687650357837907, −12.55739073211365325270351110426, −10.90688971819396908427962290649, −9.861699330454545896144633258837, −9.075819972951797168109004308740, −7.934588239070196815668820996202, −6.98574727853978141385258619901, −5.18793584489183445169454563101, −2.83241364847034310728273070018, −2.27467957173551348400212843428, 2.50570930030457100859793225972, 4.28942061007337349723234755658, 5.67095026610952465052994156741, 7.13639326228765228597208803669, 8.478662222107895991395512766325, 9.121159773822369761180028860458, 10.00416175379147466373000202835, 10.93330090014668084864014193625, 13.12997450726210219647039255426, 13.56798024031238557961923177213

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