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Properties

Label 2-130-65.23-c2-0-13
Degree $2$
Conductor $130$
Sign $-0.973 + 0.229i$
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  + (0.366 − 1.36i)2-s + (1.39 − 5.21i)3-s + (−1.73 − i)4-s + (−4.51 + 2.15i)5-s + (−6.60 − 3.81i)6-s + (2.83 − 0.760i)7-s + (−2 + 1.99i)8-s + (−17.4 − 10.0i)9-s + (1.29 + 6.95i)10-s + (16.9 − 9.78i)11-s + (−7.63 + 7.63i)12-s + (−10.6 + 7.48i)13-s − 4.15i·14-s + (4.93 + 26.5i)15-s + (1.99 + 3.46i)16-s + (2.15 + 8.05i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.465 − 1.73i)3-s + (−0.433 − 0.250i)4-s + (−0.902 + 0.431i)5-s + (−1.10 − 0.635i)6-s + (0.405 − 0.108i)7-s + (−0.250 + 0.249i)8-s + (−1.93 − 1.11i)9-s + (0.129 + 0.695i)10-s + (1.54 − 0.889i)11-s + (−0.635 + 0.635i)12-s + (−0.817 + 0.575i)13-s − 0.296i·14-s + (0.329 + 1.76i)15-s + (0.124 + 0.216i)16-s + (0.127 + 0.474i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(3.54224\)
Root analytic conductor: \(1.88208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.165987 - 1.42822i\)
\(L(\frac12)\) \(\approx\) \(0.165987 - 1.42822i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
5 \( 1 + (4.51 - 2.15i)T \)
13 \( 1 + (10.6 - 7.48i)T \)
good3 \( 1 + (-1.39 + 5.21i)T + (-7.79 - 4.5i)T^{2} \)
7 \( 1 + (-2.83 + 0.760i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-16.9 + 9.78i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + (-2.15 - 8.05i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (-5.74 + 9.94i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-1.68 + 6.27i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 + (-12.7 + 7.35i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + 42.7iT - 961T^{2} \)
37 \( 1 + (3.93 - 14.6i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (-22.3 + 12.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-10.8 + 2.89i)T + (1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (-28.0 - 28.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-71.4 - 71.4i)T + 2.80e3iT^{2} \)
59 \( 1 + (-44.7 + 77.4i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (42.5 - 73.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (15.6 - 58.4i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (5.25 + 3.03i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-62.8 + 62.8i)T - 5.32e3iT^{2} \)
79 \( 1 - 34.9iT - 6.24e3T^{2} \)
83 \( 1 + (46.8 - 46.8i)T - 6.88e3iT^{2} \)
89 \( 1 + (-71.5 - 124. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (173. - 46.5i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−12.35903009592425390281713751247, −11.80874652926878787171781566428, −11.10734170219746176869407969947, −9.195960406068516125973903484261, −8.217969514597953377896130997991, −7.21240966997205425223308129408, −6.22205768841379726143008427567, −4.02217876919243342202380006859, −2.58882111853009518896487224432, −0.972812951180582526566040913766, 3.47356966706079544539981450806, 4.46194210119526227554499554043, 5.23263315304967384889987592193, 7.23410990013612824625404437855, 8.438644667917930466232331907420, 9.245338023786462635727688361667, 10.14392299517159650975862022228, 11.53871906587081212723255381722, 12.38223338686011297088961619351, 14.18650532546717544955223229333

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