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Properties

Label 2-130-65.24-c2-0-11
Degree $2$
Conductor $130$
Sign $-0.815 + 0.578i$
Analytic cond. $3.54224$
Root an. cond. $1.88208$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 + 0.366i)2-s + (−4.60 + 2.65i)3-s + (1.73 + i)4-s + (−0.946 − 4.90i)5-s + (−7.26 + 1.94i)6-s + (−9.63 + 2.58i)7-s + (1.99 + 2i)8-s + (9.63 − 16.6i)9-s + (0.503 − 7.05i)10-s + (−9.26 − 2.48i)11-s − 10.6·12-s + (−1.62 − 12.8i)13-s − 14.1·14-s + (17.4 + 20.0i)15-s + (1.99 + 3.46i)16-s + (−9.93 + 17.2i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−1.53 + 0.886i)3-s + (0.433 + 0.250i)4-s + (−0.189 − 0.981i)5-s + (−1.21 + 0.324i)6-s + (−1.37 + 0.368i)7-s + (0.249 + 0.250i)8-s + (1.07 − 1.85i)9-s + (0.0503 − 0.705i)10-s + (−0.842 − 0.225i)11-s − 0.886·12-s + (−0.125 − 0.992i)13-s − 1.00·14-s + (1.16 + 1.33i)15-s + (0.124 + 0.216i)16-s + (−0.584 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $-0.815 + 0.578i$
Analytic conductor: \(3.54224\)
Root analytic conductor: \(1.88208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1),\ -0.815 + 0.578i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00199086 - 0.00625363i\)
\(L(\frac12)\) \(\approx\) \(0.00199086 - 0.00625363i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
5 \( 1 + (0.946 + 4.90i)T \)
13 \( 1 + (1.62 + 12.8i)T \)
good3 \( 1 + (4.60 - 2.65i)T + (4.5 - 7.79i)T^{2} \)
7 \( 1 + (9.63 - 2.58i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (9.26 + 2.48i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (9.93 - 17.2i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (13.0 - 3.49i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (-2.81 - 4.87i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-13.1 - 22.7i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-1.63 - 1.63i)T + 961iT^{2} \)
37 \( 1 + (5.60 - 20.9i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (14.1 - 52.9i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-29.9 + 51.8i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (62.7 + 62.7i)T + 2.20e3iT^{2} \)
53 \( 1 + 44.6iT - 2.80e3T^{2} \)
59 \( 1 + (5.87 + 21.9i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-10.5 + 18.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-66.0 - 17.6i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (12.9 - 3.46i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (82.1 + 82.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 27.2T + 6.24e3T^{2} \)
83 \( 1 + (35.7 - 35.7i)T - 6.88e3iT^{2} \)
89 \( 1 + (142. + 38.1i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (-30.2 - 112. i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−12.82823849419358495015580138545, −11.81982946950035745687904310544, −10.64337110463606326649468126100, −9.892513188117412548653609526615, −8.454474503025284895401273739243, −6.56290794438231659754004578699, −5.68530915126847445841994833356, −4.87945616488935201026506499932, −3.55542836680623693933345525281, −0.00400541481885732508697112383, 2.55734824510443092851693385533, 4.46545490933140708794352514359, 5.99244484594050309418484370944, 6.73261780107022283268008678177, 7.33935729323818188427509690519, 9.842660471078887553793919993631, 10.80723768728067115265352743814, 11.50454904567165433752843148720, 12.50149906004157950604232291899, 13.20740131630976578428745195043

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