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Properties

Label 2-430-43.36-c1-0-13
Degree $2$
Conductor $430$
Sign $0.288 + 0.957i$
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  + 2-s + (−1.04 − 1.80i)3-s + 4-s + (0.5 + 0.866i)5-s + (−1.04 − 1.80i)6-s + (0.324 − 0.561i)7-s + 8-s + (−0.675 + 1.17i)9-s + (0.5 + 0.866i)10-s + 1.08·11-s + (−1.04 − 1.80i)12-s + (2.89 − 5.01i)13-s + (0.324 − 0.561i)14-s + (1.04 − 1.80i)15-s + 16-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.602 − 1.04i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (−0.425 − 0.737i)6-s + (0.122 − 0.212i)7-s + 0.353·8-s + (−0.225 + 0.390i)9-s + (0.158 + 0.273i)10-s + 0.327·11-s + (−0.301 − 0.521i)12-s + (0.802 − 1.39i)13-s + (0.0866 − 0.149i)14-s + (0.269 − 0.466i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.288 + 0.957i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.288 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49016 - 1.10698i\)
\(L(\frac12)\) \(\approx\) \(1.49016 - 1.10698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-6.53 + 0.486i)T \)
good3 \( 1 + (1.04 + 1.80i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-0.324 + 0.561i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.08T + 11T^{2} \)
13 \( 1 + (-2.89 + 5.01i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.54 + 2.67i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.67 + 2.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.45 - 7.71i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.219 + 0.379i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.45 - 2.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.87T + 41T^{2} \)
47 \( 1 - 8.04T + 47T^{2} \)
53 \( 1 + (-4.43 - 7.68i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.46T + 59T^{2} \)
61 \( 1 + (0.762 - 1.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.74 + 6.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.13 - 1.96i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.51 - 4.35i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.87 - 11.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.15 + 3.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.75 - 8.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.34T + 97T^{2} \)
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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

−10.95485024020538363182462876077, −10.64585712378552272102054820381, −9.114593010715685541793435984758, −7.82129748912754312953023272585, −7.06981091250209891794629309235, −6.15979985072145556180300189204, −5.53905836304732221926736917593, −4.05446938156395234927301550621, −2.71316818329191147597534231055, −1.14970249217538670475520163172, 1.94346083910727974273007225082, 3.92268259515818134124451734054, 4.34677871638965689394717078879, 5.60768209794256463127293664456, 6.15239911040345096435991241676, 7.53522353832097052228703233820, 8.884685749806539575033141678979, 9.632836367116910588107729376834, 10.61031119597733563414572326720, 11.44649957854197422047504270730

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