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Properties

Label 2-430-215.42-c1-0-9
Degree $2$
Conductor $430$
Sign $0.585 - 0.810i$
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  + (−0.707 + 0.707i)2-s + (2.10 + 2.10i)3-s − 1.00i·4-s + (−1.02 − 1.98i)5-s − 2.97·6-s + (3.26 − 3.26i)7-s + (0.707 + 0.707i)8-s + 5.85i·9-s + (2.13 + 0.675i)10-s + 2.00·11-s + (2.10 − 2.10i)12-s + (−3.38 + 3.38i)13-s + 4.61i·14-s + (2.00 − 6.34i)15-s − 1.00·16-s + (0.281 + 0.281i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (1.21 + 1.21i)3-s − 0.500i·4-s + (−0.460 − 0.887i)5-s − 1.21·6-s + (1.23 − 1.23i)7-s + (0.250 + 0.250i)8-s + 1.95i·9-s + (0.674 + 0.213i)10-s + 0.605·11-s + (0.607 − 0.607i)12-s + (−0.938 + 0.938i)13-s + 1.23i·14-s + (0.518 − 1.63i)15-s − 0.250·16-s + (0.0683 + 0.0683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.585 - 0.810i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.585 - 0.810i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49709 + 0.765041i\)
\(L(\frac12)\) \(\approx\) \(1.49709 + 0.765041i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.02 + 1.98i)T \)
43 \( 1 + (5.02 - 4.21i)T \)
good3 \( 1 + (-2.10 - 2.10i)T + 3iT^{2} \)
7 \( 1 + (-3.26 + 3.26i)T - 7iT^{2} \)
11 \( 1 - 2.00T + 11T^{2} \)
13 \( 1 + (3.38 - 3.38i)T - 13iT^{2} \)
17 \( 1 + (-0.281 - 0.281i)T + 17iT^{2} \)
19 \( 1 - 8.17T + 19T^{2} \)
23 \( 1 + (-2.02 + 2.02i)T - 23iT^{2} \)
29 \( 1 + 6.79T + 29T^{2} \)
31 \( 1 - 5.21T + 31T^{2} \)
37 \( 1 + (0.298 - 0.298i)T - 37iT^{2} \)
41 \( 1 + 7.03T + 41T^{2} \)
47 \( 1 + (5.34 + 5.34i)T + 47iT^{2} \)
53 \( 1 + (6.37 - 6.37i)T - 53iT^{2} \)
59 \( 1 - 7.42iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + (0.771 + 0.771i)T + 67iT^{2} \)
71 \( 1 + 3.48iT - 71T^{2} \)
73 \( 1 + (-2.57 - 2.57i)T + 73iT^{2} \)
79 \( 1 + 6.91iT - 79T^{2} \)
83 \( 1 + (-0.858 + 0.858i)T - 83iT^{2} \)
89 \( 1 + 8.21T + 89T^{2} \)
97 \( 1 + (2.39 + 2.39i)T + 97iT^{2} \)
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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

−11.12277109338091420943532840040, −9.939989415693486741090248609797, −9.457962003787424139538646582358, −8.580973613820710763121597027910, −7.83897945493988327950284337207, −7.18192780202133877360446683431, −4.98791229661824182313190602248, −4.59923303100172322024480108646, −3.56313177495421868016510759452, −1.53774712413729557845380621391, 1.54777856164943091434540913784, 2.63294766647125906182580653854, 3.35572040481896342467547307274, 5.29403206308378976724722941051, 6.84198934281425416973167449723, 7.72845799007571105970729693173, 8.065074214600897741171781961675, 9.067496812917756981618290854822, 9.898970513264130781886987401765, 11.42528695266536833001685640826

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