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Properties

Label 2-130-13.10-c1-0-5
Degree $2$
Conductor $130$
Sign $-0.515 + 0.856i$
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 + (−1.66 − 2.88i)3-s + (0.499 − 0.866i)4-s + i·5-s + (−2.88 − 1.66i)6-s + (−1.24 − 0.719i)7-s − 0.999i·8-s + (−4.05 + 7.01i)9-s + (0.5 + 0.866i)10-s + (3.49 − 2.01i)11-s − 3.33·12-s + (3.13 − 1.78i)13-s − 1.43·14-s + (2.88 − 1.66i)15-s + (−0.5 − 0.866i)16-s + (1.15 − 1.99i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.961 − 1.66i)3-s + (0.249 − 0.433i)4-s + 0.447i·5-s + (−1.17 − 0.680i)6-s + (−0.471 − 0.272i)7-s − 0.353i·8-s + (−1.35 + 2.33i)9-s + (0.158 + 0.273i)10-s + (1.05 − 0.608i)11-s − 0.961·12-s + (0.868 − 0.495i)13-s − 0.384·14-s + (0.745 − 0.430i)15-s + (−0.125 − 0.216i)16-s + (0.279 − 0.484i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $-0.515 + 0.856i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ -0.515 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.538248 - 0.952303i\)
\(L(\frac12)\) \(\approx\) \(0.538248 - 0.952303i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-3.13 + 1.78i)T \)
good3 \( 1 + (1.66 + 2.88i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.24 + 0.719i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.49 + 2.01i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.15 + 1.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.346 + 0.199i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.780 - 1.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.93 - 6.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.10iT - 31T^{2} \)
37 \( 1 + (5.40 - 3.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.659 - 0.380i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.50 - 9.53i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.84iT - 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + (3 + 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.84 + 4.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.00 - 5.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 - 0.931T + 79T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + (0.840 - 0.485i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.6 + 6.15i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.97577231599493403762942748667, −11.96374196664593914682168389146, −11.33868980372545330125359285126, −10.38905002894658620941662357202, −8.468340929311502172593861087918, −6.98442816382278645504578275604, −6.45137700394218682783169569875, −5.36743070462288734462690289887, −3.21010994498602529581397402631, −1.25226955980514058380874011786, 3.72554087141003677071150751402, 4.48136381686395513272588938901, 5.74464083658484385578003554792, 6.53596856065002117781923089661, 8.716499906633137429997657150177, 9.551988941146278002232745272440, 10.62236846871735297565625374357, 11.76109376493727864742600967444, 12.30341421492182551390902382083, 13.82809116600982599649935158090

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