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Properties

Label 2-430-43.23-c1-0-2
Degree $2$
Conductor $430$
Sign $-0.939 - 0.342i$
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 + (1.11 + 2.83i)3-s + (−0.222 + 0.974i)4-s + (−0.0747 + 0.997i)5-s + (−1.52 + 2.63i)6-s + (−2.19 − 3.80i)7-s + (−0.900 + 0.433i)8-s + (−4.59 + 4.26i)9-s + (−0.826 + 0.563i)10-s + (0.332 + 1.45i)11-s + (−3.01 + 0.453i)12-s + (0.626 + 0.427i)13-s + (1.60 − 4.09i)14-s + (−2.90 + 0.897i)15-s + (−0.900 − 0.433i)16-s + (0.574 + 7.66i)17-s + ⋯
L(s)  = 1  + (0.440 + 0.552i)2-s + (0.642 + 1.63i)3-s + (−0.111 + 0.487i)4-s + (−0.0334 + 0.445i)5-s + (−0.621 + 1.07i)6-s + (−0.830 − 1.43i)7-s + (−0.318 + 0.153i)8-s + (−1.53 + 1.42i)9-s + (−0.261 + 0.178i)10-s + (0.100 + 0.439i)11-s + (−0.869 + 0.131i)12-s + (0.173 + 0.118i)13-s + (0.429 − 1.09i)14-s + (−0.751 + 0.231i)15-s + (−0.225 − 0.108i)16-s + (0.139 + 1.85i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.314976 + 1.78231i\)
\(L(\frac12)\) \(\approx\) \(0.314976 + 1.78231i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.55 + 0.143i)T \)
good3 \( 1 + (-1.11 - 2.83i)T + (-2.19 + 2.04i)T^{2} \)
7 \( 1 + (2.19 + 3.80i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.332 - 1.45i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.626 - 0.427i)T + (4.74 + 12.1i)T^{2} \)
17 \( 1 + (-0.574 - 7.66i)T + (-16.8 + 2.53i)T^{2} \)
19 \( 1 + (1.20 + 1.11i)T + (1.41 + 18.9i)T^{2} \)
23 \( 1 + (-6.92 - 2.13i)T + (19.0 + 12.9i)T^{2} \)
29 \( 1 + (-1.27 + 3.24i)T + (-21.2 - 19.7i)T^{2} \)
31 \( 1 + (-3.92 + 0.591i)T + (29.6 - 9.13i)T^{2} \)
37 \( 1 + (-5.28 + 9.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.19 - 1.49i)T + (-9.12 + 39.9i)T^{2} \)
47 \( 1 + (-1.12 + 4.91i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (3.11 - 2.12i)T + (19.3 - 49.3i)T^{2} \)
59 \( 1 + (-8.29 - 3.99i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (11.6 + 1.76i)T + (58.2 + 17.9i)T^{2} \)
67 \( 1 + (-8.84 - 8.20i)T + (5.00 + 66.8i)T^{2} \)
71 \( 1 + (3.35 - 1.03i)T + (58.6 - 39.9i)T^{2} \)
73 \( 1 + (5.05 + 3.44i)T + (26.6 + 67.9i)T^{2} \)
79 \( 1 + (-3.54 - 6.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.20 + 8.16i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (2.17 + 5.53i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (4.09 + 17.9i)T + (-87.3 + 42.0i)T^{2} \)
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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

−11.12126472621714206509204006959, −10.52502373557273246266986728251, −9.827110625860008193938320441701, −8.962923640271946746016439945623, −7.88542470213248791088902515088, −6.89889748708926058512433060053, −5.80139853862705177521993243705, −4.29831178161698308673269193019, −3.98265045424657491094522868031, −2.95080786520931695316916649482, 0.984968537147020556091767463892, 2.59654319667322105441649762080, 3.03290901029340197579499616422, 5.07529786065158691696226844516, 6.12977844626831049718261492724, 6.86600889858500555636041827659, 8.124612714611225548342114970806, 8.969741990570677283421164664362, 9.501244448591419144828115983603, 11.21871033468129882033864771183

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