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Properties

Label 2-130-65.9-c3-0-13
Degree $2$
Conductor $130$
Sign $0.997 - 0.0712i$
Analytic cond. $7.67024$
Root an. cond. $2.76952$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 − i)2-s + (6.95 + 4.01i)3-s + (1.99 + 3.46i)4-s + (10.8 − 2.71i)5-s + (−8.03 − 13.9i)6-s + (8.29 − 4.78i)7-s − 7.99i·8-s + (18.7 + 32.5i)9-s + (−21.5 − 6.14i)10-s + (34.0 − 58.9i)11-s + 32.1i·12-s + (−38.2 + 27.1i)13-s − 19.1·14-s + (86.3 + 24.6i)15-s + (−8 + 13.8i)16-s + (−87.2 + 50.3i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (1.33 + 0.773i)3-s + (0.249 + 0.433i)4-s + (0.970 − 0.242i)5-s + (−0.546 − 0.946i)6-s + (0.447 − 0.258i)7-s − 0.353i·8-s + (0.695 + 1.20i)9-s + (−0.679 − 0.194i)10-s + (0.932 − 1.61i)11-s + 0.773i·12-s + (−0.815 + 0.579i)13-s − 0.365·14-s + (1.48 + 0.424i)15-s + (−0.125 + 0.216i)16-s + (−1.24 + 0.718i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0712i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.997 - 0.0712i$
Analytic conductor: \(7.67024\)
Root analytic conductor: \(2.76952\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :3/2),\ 0.997 - 0.0712i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.24303 + 0.0799708i\)
\(L(\frac12)\) \(\approx\) \(2.24303 + 0.0799708i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + i)T \)
5 \( 1 + (-10.8 + 2.71i)T \)
13 \( 1 + (38.2 - 27.1i)T \)
good3 \( 1 + (-6.95 - 4.01i)T + (13.5 + 23.3i)T^{2} \)
7 \( 1 + (-8.29 + 4.78i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-34.0 + 58.9i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (87.2 - 50.3i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-30.1 - 52.1i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-56.6 - 32.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-64.7 + 112. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 78.5T + 2.97e4T^{2} \)
37 \( 1 + (74.1 + 42.7i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (183. - 318. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-5.24 + 3.02i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 422. iT - 1.03e5T^{2} \)
53 \( 1 + 114. iT - 1.48e5T^{2} \)
59 \( 1 + (196. + 340. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (212. + 367. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (894. + 516. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-391. - 677. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 244. iT - 3.89e5T^{2} \)
79 \( 1 + 1.13e3T + 4.93e5T^{2} \)
83 \( 1 + 215. iT - 5.71e5T^{2} \)
89 \( 1 + (-554. + 961. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-1.28e3 + 744. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.11169752178108758148814464519, −11.51905394835090750266816410692, −10.50003361676995241984631791127, −9.424027233886644972432631288057, −8.930512217088180814425394306978, −8.019601960266751759487268417754, −6.32594675942325769932049326238, −4.44938523887537368456176432563, −3.12115773659390235839697653798, −1.68951626410305315018801362764, 1.70460396272150026411543553647, 2.59824305510367577050302291977, 4.97375825491645972328557408939, 6.90948255010803339097460601603, 7.22859164694206806401512939564, 8.789435120708459680063797193622, 9.272491573730358136157973726650, 10.34542532209799874802979558864, 11.97149260945344974619291448953, 13.06971844133096206144872144643

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