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Properties

Label 2-130-65.9-c3-0-9
Degree $2$
Conductor $130$
Sign $0.0655 - 0.997i$
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 + (5.30 + 3.05i)3-s + (1.99 + 3.46i)4-s + (6.73 + 8.92i)5-s + (6.11 + 10.6i)6-s + (−12.2 + 7.04i)7-s + 7.99i·8-s + (5.22 + 9.05i)9-s + (2.73 + 22.1i)10-s + (15.1 − 26.2i)11-s + 24.4i·12-s + (−46.7 + 3.74i)13-s − 28.1·14-s + (8.37 + 67.9i)15-s + (−8 + 13.8i)16-s + (102. − 59.2i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (1.01 + 0.588i)3-s + (0.249 + 0.433i)4-s + (0.602 + 0.798i)5-s + (0.416 + 0.721i)6-s + (−0.658 + 0.380i)7-s + 0.353i·8-s + (0.193 + 0.335i)9-s + (0.0865 + 0.701i)10-s + (0.415 − 0.718i)11-s + 0.588i·12-s + (−0.996 + 0.0799i)13-s − 0.537·14-s + (0.144 + 1.16i)15-s + (−0.125 + 0.216i)16-s + (1.46 − 0.845i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0655 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0655 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.0655 - 0.997i$
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: \((2,\ 130,\ (\ :3/2),\ 0.0655 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.30069 + 2.15460i\)
\(L(\frac12)\) \(\approx\) \(2.30069 + 2.15460i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.73 - 8.92i)T \)
13 \( 1 + (46.7 - 3.74i)T \)
good3 \( 1 + (-5.30 - 3.05i)T + (13.5 + 23.3i)T^{2} \)
7 \( 1 + (12.2 - 7.04i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-15.1 + 26.2i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-102. + 59.2i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (42.4 + 73.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-136. - 78.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (78.2 - 135. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 117.T + 2.97e4T^{2} \)
37 \( 1 + (-10.5 - 6.06i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-177. + 306. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (46.2 - 26.6i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 371. iT - 1.03e5T^{2} \)
53 \( 1 + 548. iT - 1.48e5T^{2} \)
59 \( 1 + (213. + 369. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (136. + 236. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (348. + 201. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (251. + 436. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 780. iT - 3.89e5T^{2} \)
79 \( 1 - 4.34T + 4.93e5T^{2} \)
83 \( 1 - 405. iT - 5.71e5T^{2} \)
89 \( 1 + (596. - 1.03e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (1.40e3 - 811. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.45309764041815324427829502055, −12.26218377222286054028237501391, −10.99907820452552326716273040110, −9.636396201630656101447553632617, −9.120154632843168011769615725660, −7.56493037543502252725810762124, −6.45756423135100068229120371229, −5.17897300690614440216582758338, −3.37527836736654186615176487495, −2.76222397645250614106106174474, 1.45180110809035839876693472331, 2.79095541231632437231161315353, 4.31623158621328187118715193032, 5.79578915732590980188080292159, 7.16062930411227409235326950100, 8.318091039781387599337871085462, 9.579134190221954697023521413021, 10.26051715671045384959155343198, 12.15291890127489435513019276455, 12.74810274198449567053924660809

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