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Properties

Label 2-130-65.62-c2-0-2
Degree $2$
Conductor $130$
Sign $0.852 + 0.522i$
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 + (−4.18 − 1.12i)3-s + (1.73 + i)4-s + (−2.95 + 4.03i)5-s + (5.31 + 3.06i)6-s + (0.174 + 0.652i)7-s + (−1.99 − 2i)8-s + (8.48 + 4.90i)9-s + (5.51 − 4.42i)10-s + (17.5 − 10.1i)11-s + (−6.13 − 6.13i)12-s + (−12.7 − 2.54i)13-s − 0.955i·14-s + (16.9 − 13.5i)15-s + (1.99 + 3.46i)16-s + (19.6 − 5.27i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−1.39 − 0.374i)3-s + (0.433 + 0.250i)4-s + (−0.590 + 0.806i)5-s + (0.885 + 0.511i)6-s + (0.0249 + 0.0932i)7-s + (−0.249 − 0.250i)8-s + (0.943 + 0.544i)9-s + (0.551 − 0.442i)10-s + (1.59 − 0.918i)11-s + (−0.511 − 0.511i)12-s + (−0.980 − 0.195i)13-s − 0.0682i·14-s + (1.12 − 0.905i)15-s + (0.124 + 0.216i)16-s + (1.15 − 0.310i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.852 + 0.522i$
Analytic conductor: \(3.54224\)
Root analytic conductor: \(1.88208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1),\ 0.852 + 0.522i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.565186 - 0.159422i\)
\(L(\frac12)\) \(\approx\) \(0.565186 - 0.159422i\)
\(L(2)\) 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 \( 1 + (2.95 - 4.03i)T \)
13 \( 1 + (12.7 + 2.54i)T \)
good3 \( 1 + (4.18 + 1.12i)T + (7.79 + 4.5i)T^{2} \)
7 \( 1 + (-0.174 - 0.652i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-17.5 + 10.1i)T + (60.5 - 104. i)T^{2} \)
17 \( 1 + (-19.6 + 5.27i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-4.90 + 8.49i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-34.7 - 9.29i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + (-7.57 + 4.37i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 - 50.2iT - 961T^{2} \)
37 \( 1 + (4.96 + 1.32i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (15.8 - 9.15i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (13.5 + 50.4i)T + (-1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (-27.7 + 27.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (0.945 - 0.945i)T - 2.80e3iT^{2} \)
59 \( 1 + (-23.3 + 40.4i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (16.7 - 29.0i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-86.8 - 23.2i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (3.19 + 1.84i)T + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (-41.8 - 41.8i)T + 5.32e3iT^{2} \)
79 \( 1 + 115. iT - 6.24e3T^{2} \)
83 \( 1 + (3.31 + 3.31i)T + 6.88e3iT^{2} \)
89 \( 1 + (-18.0 - 31.1i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (1.76 + 6.57i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−12.31647406631558685653210098025, −11.79070149057090666737980130779, −11.11791715287890380227037305702, −10.15206110282771414304147557862, −8.798207581380213895644201577551, −7.23265177697021414638545662335, −6.68388716731517296106071564364, −5.32164283062430761656087899994, −3.32010496823456489654392001005, −0.837873989563123357779744138844, 1.03550767665347233799113066923, 4.23931556264864817514851344758, 5.30268814140093835059656457876, 6.61646550773981166726945465558, 7.69452318047644477774130535075, 9.213211250904038069283307747733, 9.951880948191443129151048982427, 11.22103483921194001985417952120, 12.00748789730725146673209257198, 12.52324847945677005511396090908

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