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Properties

Label 2-130-65.37-c1-0-0
Degree $2$
Conductor $130$
Sign $-0.989 - 0.147i$
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 + (−0.935 − 0.250i)3-s + (0.499 − 0.866i)4-s + (−2.23 + 0.0997i)5-s + (0.935 − 0.250i)6-s + (−1.88 + 3.26i)7-s + 0.999i·8-s + (−1.78 − 1.03i)9-s + (1.88 − 1.20i)10-s + (−1.91 − 0.513i)11-s + (−0.685 + 0.685i)12-s + (−0.930 + 3.48i)13-s − 3.76i·14-s + (2.11 + 0.466i)15-s + (−0.5 − 0.866i)16-s + (−1.33 − 4.97i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.540 − 0.144i)3-s + (0.249 − 0.433i)4-s + (−0.999 + 0.0446i)5-s + (0.382 − 0.102i)6-s + (−0.712 + 1.23i)7-s + 0.353i·8-s + (−0.594 − 0.343i)9-s + (0.595 − 0.380i)10-s + (−0.577 − 0.154i)11-s + (−0.197 + 0.197i)12-s + (−0.258 + 0.966i)13-s − 1.00i·14-s + (0.546 + 0.120i)15-s + (−0.125 − 0.216i)16-s + (−0.323 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $-0.989 - 0.147i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ -0.989 - 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0117033 + 0.158019i\)
\(L(\frac12)\) \(\approx\) \(0.0117033 + 0.158019i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0997i)T \)
13 \( 1 + (0.930 - 3.48i)T \)
good3 \( 1 + (0.935 + 0.250i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.88 - 3.26i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.91 + 0.513i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.33 + 4.97i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.103 - 0.385i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.652 - 2.43i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.00 - 0.581i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.10 - 7.10i)T + 31iT^{2} \)
37 \( 1 + (5.39 + 9.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0903 - 0.337i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.86 - 1.30i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 2.16T + 47T^{2} \)
53 \( 1 + (4.52 - 4.52i)T - 53iT^{2} \)
59 \( 1 + (12.2 - 3.28i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.985 + 1.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.0 - 6.95i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.875 - 0.234i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 8.38iT - 73T^{2} \)
79 \( 1 - 3.10iT - 79T^{2} \)
83 \( 1 - 4.98T + 83T^{2} \)
89 \( 1 + (4.22 - 15.7i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-14.9 - 8.65i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.92856136568292371978450669995, −12.26827818916163137057095987627, −11.86620279415342256663002814902, −10.84879260711713806564884399634, −9.334694391820543733839124884792, −8.657520973929874777809534366154, −7.28874804465564844541085724035, −6.27456561411436564402999719499, −5.06497398891536652690207863497, −2.93137204397670265916986011849, 0.20223573074322255795982390145, 3.18935265302090773214112512821, 4.57205163501048781453534076538, 6.36214650811650265975540021096, 7.66216499769354372069909234776, 8.389099185970681716878138065782, 10.17041058430327047453795712092, 10.58644381294010362280016267523, 11.60649380162224183368769150908, 12.65478452924844189663136426272

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