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Properties

Label 2-430-43.35-c1-0-2
Degree $2$
Conductor $430$
Sign $0.927 + 0.372i$
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  + (0.222 − 0.974i)2-s + (0.187 + 0.823i)3-s + (−0.900 − 0.433i)4-s + (−0.623 + 0.781i)5-s + 0.844·6-s + 0.703·7-s + (−0.623 + 0.781i)8-s + (2.06 − 0.992i)9-s + (0.623 + 0.781i)10-s + (−1.48 + 0.716i)11-s + (0.187 − 0.823i)12-s + (3.92 − 4.92i)13-s + (0.156 − 0.686i)14-s + (−0.760 − 0.366i)15-s + (0.623 + 0.781i)16-s + (4.70 + 5.89i)17-s + ⋯
L(s)  = 1  + (0.157 − 0.689i)2-s + (0.108 + 0.475i)3-s + (−0.450 − 0.216i)4-s + (−0.278 + 0.349i)5-s + 0.344·6-s + 0.265·7-s + (−0.220 + 0.276i)8-s + (0.686 − 0.330i)9-s + (0.197 + 0.247i)10-s + (−0.448 + 0.216i)11-s + (0.0542 − 0.237i)12-s + (1.08 − 1.36i)13-s + (0.0418 − 0.183i)14-s + (−0.196 − 0.0946i)15-s + (0.155 + 0.195i)16-s + (1.14 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.927 + 0.372i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.927 + 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56095 - 0.301789i\)
\(L(\frac12)\) \(\approx\) \(1.56095 - 0.301789i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.222 + 0.974i)T \)
5 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (6.16 + 2.24i)T \)
good3 \( 1 + (-0.187 - 0.823i)T + (-2.70 + 1.30i)T^{2} \)
7 \( 1 - 0.703T + 7T^{2} \)
11 \( 1 + (1.48 - 0.716i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-3.92 + 4.92i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-4.70 - 5.89i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + (-6.26 - 3.01i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (2.46 - 1.18i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-1.34 + 5.87i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-0.654 + 2.86i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + 5.65T + 37T^{2} \)
41 \( 1 + (0.950 - 4.16i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (2.54 + 1.22i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-5.84 - 7.33i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (6.87 + 8.62i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (1.85 + 8.13i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-5.87 - 2.82i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (2.14 + 1.03i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-3.99 + 5.00i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 + (-3.87 - 16.9i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (1.37 + 6.01i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (13.5 - 6.54i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.96382851497789720100284066275, −10.15684462267648014946376331027, −9.806206061482445417312410810668, −8.263807305026462275550566762289, −7.76810076311445141107120516185, −6.14310970898713081572688334028, −5.21140606217048166872250115052, −3.82080892027179637693242290351, −3.28315214040127415049142738389, −1.38772351494317809645270857317, 1.32727655867864527811643336996, 3.26772303844197334034246874912, 4.62715516185023492087280196824, 5.40932763660864484988214192547, 6.82467666732854728927525245932, 7.36471575492663735556468658723, 8.331294354882635404314038151805, 9.160668422334636481494093710346, 10.18007826241227533310807267224, 11.49873112012199792460975994261

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