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Properties

Label 2-130-65.28-c1-0-0
Degree $2$
Conductor $130$
Sign $-0.707 - 0.706i$
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 + (−0.776 + 2.89i)3-s + (0.499 − 0.866i)4-s + (2.12 + 0.709i)5-s + (−0.776 − 2.89i)6-s + (−0.638 + 1.10i)7-s + 0.999i·8-s + (−5.18 − 2.99i)9-s + (−2.19 + 0.445i)10-s + (−0.228 + 0.854i)11-s + (2.12 + 2.12i)12-s + (−3.31 − 1.42i)13-s − 1.27i·14-s + (−3.70 + 5.59i)15-s + (−0.5 − 0.866i)16-s + (0.599 − 0.160i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.448 + 1.67i)3-s + (0.249 − 0.433i)4-s + (0.948 + 0.317i)5-s + (−0.316 − 1.18i)6-s + (−0.241 + 0.418i)7-s + 0.353i·8-s + (−1.72 − 0.998i)9-s + (−0.692 + 0.140i)10-s + (−0.0690 + 0.257i)11-s + (0.612 + 0.612i)12-s + (−0.918 − 0.395i)13-s − 0.341i·14-s + (−0.955 + 1.44i)15-s + (−0.125 − 0.216i)16-s + (0.145 − 0.0389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $-0.707 - 0.706i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ -0.707 - 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.289107 + 0.698320i\)
\(L(\frac12)\) \(\approx\) \(0.289107 + 0.698320i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 - 0.709i)T \)
13 \( 1 + (3.31 + 1.42i)T \)
good3 \( 1 + (0.776 - 2.89i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (0.638 - 1.10i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.228 - 0.854i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.599 + 0.160i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.26 + 0.874i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-8.07 - 2.16i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-5.53 + 3.19i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (7.30 - 7.30i)T - 31iT^{2} \)
37 \( 1 + (4.58 + 7.93i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.36 - 2.24i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.528 + 1.97i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 2.55T + 47T^{2} \)
53 \( 1 + (-2.67 - 2.67i)T + 53iT^{2} \)
59 \( 1 + (0.288 + 1.07i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.53 + 6.12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.23 + 1.29i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.80 + 6.72i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 4.86iT - 73T^{2} \)
79 \( 1 + 11.2iT - 79T^{2} \)
83 \( 1 + 1.75T + 83T^{2} \)
89 \( 1 + (6.08 + 1.63i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.64 + 4.99i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.15729118393051047717671097370, −12.51802497858205883936222502257, −11.11683051580999074690427121433, −10.38487803297575729842335978258, −9.525554042916716853873002910074, −9.025136484181999875843913369634, −7.12700783035901198369532682933, −5.67673704109954691221954632592, −4.99890818283639070088238985832, −2.97710013482101789024940129080, 1.10453740983235225640494206554, 2.56550957275573053569205643755, 5.33621589342291419592723229548, 6.64825841016821502990649491806, 7.36390532828284021756970120775, 8.641018860609953906236496279385, 9.791407922334853018395524221023, 11.00081222376853459743585489987, 12.05733563484766975383505419317, 12.86230577235158570841209672667

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