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Properties

Label 2-130-65.7-c1-0-4
Degree $2$
Conductor $130$
Sign $0.980 - 0.197i$
Analytic cond. $1.03805$
Root an. cond. $1.01884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.316 − 1.18i)3-s + (0.499 + 0.866i)4-s + (1.44 + 1.70i)5-s + (0.316 − 1.18i)6-s + (−0.401 − 0.695i)7-s + 0.999i·8-s + (1.30 − 0.752i)9-s + (0.401 + 2.19i)10-s + (−0.707 − 2.64i)11-s + (0.864 − 0.864i)12-s + (−1.91 + 3.05i)13-s − 0.803i·14-s + (1.55 − 2.24i)15-s + (−0.5 + 0.866i)16-s + (−3.64 − 0.975i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.182 − 0.682i)3-s + (0.249 + 0.433i)4-s + (0.647 + 0.762i)5-s + (0.129 − 0.482i)6-s + (−0.151 − 0.262i)7-s + 0.353i·8-s + (0.434 − 0.250i)9-s + (0.127 + 0.695i)10-s + (−0.213 − 0.796i)11-s + (0.249 − 0.249i)12-s + (−0.532 + 0.846i)13-s − 0.214i·14-s + (0.401 − 0.580i)15-s + (−0.125 + 0.216i)16-s + (−0.883 − 0.236i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(130\)    =    \(2 \cdot 5 \cdot 13\)
Sign: $0.980 - 0.197i$
Analytic conductor: \(1.03805\)
Root analytic conductor: \(1.01884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{130} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 130,\ (\ :1/2),\ 0.980 - 0.197i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46230 + 0.145552i\)
\(L(\frac12)\) \(\approx\) \(1.46230 + 0.145552i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (-1.44 - 1.70i)T \)
13 \( 1 + (1.91 - 3.05i)T \)
good3 \( 1 + (0.316 + 1.18i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (0.401 + 0.695i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.707 + 2.64i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (3.64 + 0.975i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (2.03 + 0.544i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.71 - 1.26i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (2.08 + 1.20i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.16 - 4.16i)T + 31iT^{2} \)
37 \( 1 + (-3.07 + 5.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.97 + 1.33i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.78 + 6.64i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 4.44T + 47T^{2} \)
53 \( 1 + (9.13 - 9.13i)T - 53iT^{2} \)
59 \( 1 + (2.20 - 8.21i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-7.35 - 12.7i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.44 - 4.30i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.83 + 14.3i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 1.70iT - 73T^{2} \)
79 \( 1 - 1.85iT - 79T^{2} \)
83 \( 1 + 1.38T + 83T^{2} \)
89 \( 1 + (1.87 - 0.501i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.41 + 1.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.58374440552194439695587678950, −12.52089998063323067068606260683, −11.49020018288639730678990323344, −10.40981890803217143219960963345, −9.113232765753243369672200449077, −7.49594187778506119788744982088, −6.69520064549122467539732491199, −5.84801998489869291063716085610, −4.09711608979111640512963059098, −2.32724899051925819122846576163, 2.22146072120894385260342893036, 4.28526233088161289997781883248, 5.08312489839495649767326134338, 6.29341675941765334691858665278, 8.012606071485956135448507901038, 9.585930895339279226916491894849, 10.06288359084327119267886971277, 11.20968789300809992079214441390, 12.72797357322691831692837573467, 12.84630882180326019965694301715

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