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Properties

Label 2-430-215.128-c1-0-19
Degree $2$
Conductor $430$
Sign $-0.376 + 0.926i$
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 + (1.91 − 1.91i)3-s + 1.00i·4-s + (2.21 − 0.269i)5-s − 2.70·6-s + (−1.02 − 1.02i)7-s + (0.707 − 0.707i)8-s − 4.33i·9-s + (−1.76 − 1.37i)10-s − 3.68·11-s + (1.91 + 1.91i)12-s + (−0.648 − 0.648i)13-s + 1.44i·14-s + (3.73 − 4.76i)15-s − 1.00·16-s + (2.53 − 2.53i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (1.10 − 1.10i)3-s + 0.500i·4-s + (0.992 − 0.120i)5-s − 1.10·6-s + (−0.385 − 0.385i)7-s + (0.250 − 0.250i)8-s − 1.44i·9-s + (−0.556 − 0.436i)10-s − 1.11·11-s + (0.552 + 0.552i)12-s + (−0.179 − 0.179i)13-s + 0.385i·14-s + (0.963 − 1.23i)15-s − 0.250·16-s + (0.614 − 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.376 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.376 + 0.926i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.376 + 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.916406 - 1.36203i\)
\(L(\frac12)\) \(\approx\) \(0.916406 - 1.36203i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.269i)T \)
43 \( 1 + (-4.84 - 4.41i)T \)
good3 \( 1 + (-1.91 + 1.91i)T - 3iT^{2} \)
7 \( 1 + (1.02 + 1.02i)T + 7iT^{2} \)
11 \( 1 + 3.68T + 11T^{2} \)
13 \( 1 + (0.648 + 0.648i)T + 13iT^{2} \)
17 \( 1 + (-2.53 + 2.53i)T - 17iT^{2} \)
19 \( 1 + 0.0313T + 19T^{2} \)
23 \( 1 + (-2.32 - 2.32i)T + 23iT^{2} \)
29 \( 1 - 7.48T + 29T^{2} \)
31 \( 1 + 4.11T + 31T^{2} \)
37 \( 1 + (-2.60 - 2.60i)T + 37iT^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
47 \( 1 + (6.32 - 6.32i)T - 47iT^{2} \)
53 \( 1 + (-3.09 - 3.09i)T + 53iT^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 + 7.52iT - 61T^{2} \)
67 \( 1 + (-1.07 + 1.07i)T - 67iT^{2} \)
71 \( 1 - 4.73iT - 71T^{2} \)
73 \( 1 + (-4.95 + 4.95i)T - 73iT^{2} \)
79 \( 1 - 2.04iT - 79T^{2} \)
83 \( 1 + (4.90 + 4.90i)T + 83iT^{2} \)
89 \( 1 - 9.71T + 89T^{2} \)
97 \( 1 + (10.1 - 10.1i)T - 97iT^{2} \)
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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

−10.60212440942137439279529260438, −9.874264721093064848653675784957, −9.078771520919137357942203071676, −8.149758450011802611156308354586, −7.42237949846586641104365808562, −6.51541183298137010865200492665, −5.10476400818802294859193130169, −3.17058482144292949401751114442, −2.48509780053595307837301639302, −1.18992896379900466248958495553, 2.23906075454644827907669611848, 3.24729194942103605684614461562, 4.80360833926228634769071687319, 5.67078194682530016202401781645, 6.86355392402387570815577858748, 8.168639786970724074231731420382, 8.762642680947295973587533461749, 9.671438238330818421966785445033, 10.14580786073290911943796533854, 10.82619166484349555243413268076

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