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Properties

Label 2-430-43.15-c1-0-6
Degree $2$
Conductor $430$
Sign $0.525 - 0.850i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.305 + 0.778i)3-s + (−0.222 − 0.974i)4-s + (0.0747 + 0.997i)5-s + (0.418 + 0.724i)6-s + (−1.41 + 2.45i)7-s + (−0.900 − 0.433i)8-s + (1.68 + 1.56i)9-s + (0.826 + 0.563i)10-s + (−0.376 + 1.64i)11-s + (0.827 + 0.124i)12-s + (−3.50 + 2.38i)13-s + (1.03 + 2.63i)14-s + (−0.799 − 0.246i)15-s + (−0.900 + 0.433i)16-s + (−0.236 + 3.15i)17-s + ⋯
L(s)  = 1  + (0.440 − 0.552i)2-s + (−0.176 + 0.449i)3-s + (−0.111 − 0.487i)4-s + (0.0334 + 0.445i)5-s + (0.170 + 0.295i)6-s + (−0.535 + 0.927i)7-s + (−0.318 − 0.153i)8-s + (0.562 + 0.521i)9-s + (0.261 + 0.178i)10-s + (−0.113 + 0.497i)11-s + (0.238 + 0.0359i)12-s + (−0.971 + 0.662i)13-s + (0.276 + 0.705i)14-s + (−0.206 − 0.0636i)15-s + (−0.225 + 0.108i)16-s + (−0.0572 + 0.764i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.525 - 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24189 + 0.692834i\)
\(L(\frac12)\) \(\approx\) \(1.24189 + 0.692834i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0747 - 0.997i)T \)
43 \( 1 + (-5.23 + 3.94i)T \)
good3 \( 1 + (0.305 - 0.778i)T + (-2.19 - 2.04i)T^{2} \)
7 \( 1 + (1.41 - 2.45i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.376 - 1.64i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (3.50 - 2.38i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (0.236 - 3.15i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (-4.63 + 4.30i)T + (1.41 - 18.9i)T^{2} \)
23 \( 1 + (-3.80 + 1.17i)T + (19.0 - 12.9i)T^{2} \)
29 \( 1 + (-1.01 - 2.58i)T + (-21.2 + 19.7i)T^{2} \)
31 \( 1 + (-0.136 - 0.0205i)T + (29.6 + 9.13i)T^{2} \)
37 \( 1 + (-3.54 - 6.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.452 - 0.567i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (2.30 + 10.0i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (5.31 + 3.62i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (-6.07 + 2.92i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (9.74 - 1.46i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (3.65 - 3.39i)T + (5.00 - 66.8i)T^{2} \)
71 \( 1 + (1.75 + 0.540i)T + (58.6 + 39.9i)T^{2} \)
73 \( 1 + (-6.91 + 4.71i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (-6.95 + 12.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.00 - 5.11i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (-5.10 + 13.0i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (0.555 - 2.43i)T + (-87.3 - 42.0i)T^{2} \)
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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

−11.34760885067891834275184480776, −10.40152753541105168036820359549, −9.708094260472844479876765331727, −8.976259180504633541346044101115, −7.43255047164083157808175038693, −6.53333316648692344836006016472, −5.26876309199112134952548542779, −4.56316597097318638220933312953, −3.15451926218408449512160364966, −2.09047999380403681261439675096, 0.830400674259193902523575382050, 3.06462998735453249871151143528, 4.20979416467252830374610217687, 5.35135304952621031100721623635, 6.32347554215825132943768055298, 7.39351403152273957612565562979, 7.75991494839616761787537446086, 9.314393667762792209203931235925, 9.913945530487240177131706149220, 11.16108953470851818840722849730

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