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Properties

Label 2-430-43.31-c1-0-3
Degree $2$
Conductor $430$
Sign $0.448 - 0.893i$
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  + (−0.900 + 0.433i)2-s + (0.953 − 0.650i)3-s + (0.623 − 0.781i)4-s + (0.733 + 0.680i)5-s + (−0.576 + 0.999i)6-s + (1.53 + 2.66i)7-s + (−0.222 + 0.974i)8-s + (−0.609 + 1.55i)9-s + (−0.955 − 0.294i)10-s + (1.26 + 1.58i)11-s + (0.0862 − 1.15i)12-s + (−4.22 + 1.30i)13-s + (−2.54 − 1.73i)14-s + (1.14 + 0.171i)15-s + (−0.222 − 0.974i)16-s + (−0.152 + 0.141i)17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (0.550 − 0.375i)3-s + (0.311 − 0.390i)4-s + (0.327 + 0.304i)5-s + (−0.235 + 0.407i)6-s + (0.581 + 1.00i)7-s + (−0.0786 + 0.344i)8-s + (−0.203 + 0.517i)9-s + (−0.302 − 0.0932i)10-s + (0.381 + 0.477i)11-s + (0.0248 − 0.332i)12-s + (−1.17 + 0.361i)13-s + (−0.679 − 0.463i)14-s + (0.294 + 0.0444i)15-s + (−0.0556 − 0.243i)16-s + (−0.0369 + 0.0342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.448 - 0.893i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.448 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09501 + 0.675430i\)
\(L(\frac12)\) \(\approx\) \(1.09501 + 0.675430i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (-0.733 - 0.680i)T \)
43 \( 1 + (3.95 + 5.22i)T \)
good3 \( 1 + (-0.953 + 0.650i)T + (1.09 - 2.79i)T^{2} \)
7 \( 1 + (-1.53 - 2.66i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.26 - 1.58i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (4.22 - 1.30i)T + (10.7 - 7.32i)T^{2} \)
17 \( 1 + (0.152 - 0.141i)T + (1.27 - 16.9i)T^{2} \)
19 \( 1 + (0.150 + 0.382i)T + (-13.9 + 12.9i)T^{2} \)
23 \( 1 + (-3.95 + 0.596i)T + (21.9 - 6.77i)T^{2} \)
29 \( 1 + (-1.72 - 1.17i)T + (10.5 + 26.9i)T^{2} \)
31 \( 1 + (-0.424 + 5.66i)T + (-30.6 - 4.62i)T^{2} \)
37 \( 1 + (1.17 - 2.03i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-10.1 + 4.87i)T + (25.5 - 32.0i)T^{2} \)
47 \( 1 + (3.62 - 4.54i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-7.29 - 2.25i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (-0.664 - 2.91i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (0.700 + 9.35i)T + (-60.3 + 9.09i)T^{2} \)
67 \( 1 + (-4.42 - 11.2i)T + (-49.1 + 45.5i)T^{2} \)
71 \( 1 + (2.96 + 0.446i)T + (67.8 + 20.9i)T^{2} \)
73 \( 1 + (2.37 - 0.732i)T + (60.3 - 41.1i)T^{2} \)
79 \( 1 + (7.79 + 13.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.85 + 4.67i)T + (30.3 - 77.2i)T^{2} \)
89 \( 1 + (-13.3 + 9.13i)T + (32.5 - 82.8i)T^{2} \)
97 \( 1 + (7.02 + 8.80i)T + (-21.5 + 94.5i)T^{2} \)
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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

−11.28379536674323000623330073707, −10.24396338298397956617938765725, −9.266861678803891102265518494437, −8.663773702969411596965719838258, −7.65361483817101221899726978551, −6.96681679267721259748158740639, −5.71830681598506361595776981875, −4.74708247112036405410516976425, −2.66405649998677373466365519109, −1.92184824471941314207364779824, 1.00683403468255468112784213345, 2.71822931957289530563427427652, 3.88742489503256837724007144833, 5.02291533526041777253237354046, 6.52671474882396049296553220893, 7.55588109303142799991775100522, 8.430242877520794392153085946546, 9.272256077341078175317662338175, 9.982054642236794584842100062026, 10.80746884123952681879123007763

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