↑

Properties

Label 2-430-43.9-c1-0-10
Degree $2$
Conductor $430$
Sign $0.975 - 0.221i$
Analytic cond. $3.43356$
Root an. cond. $1.85298$
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  + (−0.623 − 0.781i)2-s + (2.98 + 0.450i)3-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (−1.51 − 2.61i)6-s + (−2.05 + 3.55i)7-s + (0.900 − 0.433i)8-s + (5.85 + 1.80i)9-s + (−0.0747 − 0.997i)10-s + (−1.22 − 5.37i)11-s + (−1.10 + 2.81i)12-s + (−0.434 + 5.79i)13-s + (4.05 − 0.611i)14-s + (2.21 + 2.05i)15-s + (−0.900 − 0.433i)16-s + (2.16 − 1.47i)17-s + ⋯
L(s)  = 1  + (−0.440 − 0.552i)2-s + (1.72 + 0.260i)3-s + (−0.111 + 0.487i)4-s + (0.369 + 0.251i)5-s + (−0.616 − 1.06i)6-s + (−0.774 + 1.34i)7-s + (0.318 − 0.153i)8-s + (1.95 + 0.602i)9-s + (−0.0236 − 0.315i)10-s + (−0.370 − 1.62i)11-s + (−0.318 + 0.812i)12-s + (−0.120 + 1.60i)13-s + (1.08 − 0.163i)14-s + (0.571 + 0.530i)15-s + (−0.225 − 0.108i)16-s + (0.525 − 0.358i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.975 - 0.221i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.975 - 0.221i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89069 + 0.211779i\)
\(L(\frac12)\) \(\approx\) \(1.89069 + 0.211779i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (-0.826 - 0.563i)T \)
43 \( 1 + (6.12 - 2.35i)T \)
good3 \( 1 + (-2.98 - 0.450i)T + (2.86 + 0.884i)T^{2} \)
7 \( 1 + (2.05 - 3.55i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.22 + 5.37i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.434 - 5.79i)T + (-12.8 - 1.93i)T^{2} \)
17 \( 1 + (-2.16 + 1.47i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (-5.33 + 1.64i)T + (15.6 - 10.7i)T^{2} \)
23 \( 1 + (-1.67 + 1.55i)T + (1.71 - 22.9i)T^{2} \)
29 \( 1 + (9.13 - 1.37i)T + (27.7 - 8.54i)T^{2} \)
31 \( 1 + (-0.758 + 1.93i)T + (-22.7 - 21.0i)T^{2} \)
37 \( 1 + (0.627 + 1.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.760 - 0.953i)T + (-9.12 + 39.9i)T^{2} \)
47 \( 1 + (-2.22 + 9.75i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (0.162 + 2.16i)T + (-52.4 + 7.89i)T^{2} \)
59 \( 1 + (-5.23 - 2.52i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (4.49 + 11.4i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-5.75 + 1.77i)T + (55.3 - 37.7i)T^{2} \)
71 \( 1 + (1.17 + 1.08i)T + (5.30 + 70.8i)T^{2} \)
73 \( 1 + (-0.645 + 8.61i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (-0.279 + 0.484i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.28 - 0.646i)T + (79.3 + 24.4i)T^{2} \)
89 \( 1 + (11.4 + 1.72i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + (-1.72 - 7.56i)T + (-87.3 + 42.0i)T^{2} \)
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

−11.14175528304695486096188769406, −9.818364778930837022000094607694, −9.248530216777548500825853205257, −8.877911201350273936182953945374, −7.917495559532764384813859526131, −6.75451970961065479669714482158, −5.37482079564163381936087476053, −3.59311300862753809918264124695, −2.96730556345021426804599333196, −2.04663594697259239086740422089, 1.38900917646516961823533473858, 2.95000245294743172484868234974, 4.00364408134325912394933817131, 5.45076027656076891601681300523, 7.09857021851528138357828841612, 7.48328581567686848735475305888, 8.157869688104422290616137889831, 9.502727293580550253097719505356, 9.815030049178409956560556065082, 10.48344897422312216773816500051

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