↑

Properties

Label 2-130-65.28-c1-0-3
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

Related objects

Downloads

Learn more

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: $\chi_{130} (93, \cdot )$
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 \( 1 + (-2.41 - 1.39i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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