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Properties

Label 2-430-43.13-c1-0-5
Degree $2$
Conductor $430$
Sign $-0.0886 - 0.996i$
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 + (0.702 − 0.651i)3-s + (−0.900 − 0.433i)4-s + (0.988 + 0.149i)5-s + (0.479 + 0.830i)6-s + (−2.06 + 3.57i)7-s + (0.623 − 0.781i)8-s + (−0.155 + 2.07i)9-s + (−0.365 + 0.930i)10-s + (2.75 − 1.32i)11-s + (−0.915 + 0.282i)12-s + (0.251 + 0.641i)13-s + (−3.02 − 2.80i)14-s + (0.791 − 0.539i)15-s + (0.623 + 0.781i)16-s + (−2.99 + 0.450i)17-s + ⋯
L(s)  = 1  + (−0.157 + 0.689i)2-s + (0.405 − 0.376i)3-s + (−0.450 − 0.216i)4-s + (0.442 + 0.0666i)5-s + (0.195 + 0.338i)6-s + (−0.780 + 1.35i)7-s + (0.220 − 0.276i)8-s + (−0.0518 + 0.691i)9-s + (−0.115 + 0.294i)10-s + (0.830 − 0.399i)11-s + (−0.264 + 0.0815i)12-s + (0.0697 + 0.177i)13-s + (−0.808 − 0.750i)14-s + (0.204 − 0.139i)15-s + (0.155 + 0.195i)16-s + (−0.725 + 0.109i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0886 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0886 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.0886 - 0.996i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.0886 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.910954 + 0.995600i\)
\(L(\frac12)\) \(\approx\) \(0.910954 + 0.995600i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.988 - 0.149i)T \)
43 \( 1 + (2.57 - 6.02i)T \)
good3 \( 1 + (-0.702 + 0.651i)T + (0.224 - 2.99i)T^{2} \)
7 \( 1 + (2.06 - 3.57i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.75 + 1.32i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.251 - 0.641i)T + (-9.52 + 8.84i)T^{2} \)
17 \( 1 + (2.99 - 0.450i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (-0.314 - 4.19i)T + (-18.7 + 2.83i)T^{2} \)
23 \( 1 + (-1.90 - 1.29i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (-0.933 - 0.866i)T + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (-7.18 + 2.21i)T + (25.6 - 17.4i)T^{2} \)
37 \( 1 + (1.38 + 2.39i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.196 - 0.860i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (2.18 + 1.05i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-2.88 + 7.35i)T + (-38.8 - 36.0i)T^{2} \)
59 \( 1 + (-2.35 - 2.95i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (14.0 + 4.33i)T + (50.4 + 34.3i)T^{2} \)
67 \( 1 + (0.589 + 7.87i)T + (-66.2 + 9.98i)T^{2} \)
71 \( 1 + (-7.44 + 5.07i)T + (25.9 - 66.0i)T^{2} \)
73 \( 1 + (1.90 + 4.86i)T + (-53.5 + 49.6i)T^{2} \)
79 \( 1 + (-8.51 + 14.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.49 - 6.02i)T + (6.20 - 82.7i)T^{2} \)
89 \( 1 + (-11.4 + 10.6i)T + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (7.49 - 3.60i)T + (60.4 - 75.8i)T^{2} \)
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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

−11.47710880891428402202135865772, −10.23721449360937397211548906897, −9.257691883290412184419823860888, −8.722732832186164492350102469009, −7.81685429750540301645219326849, −6.52013239546656256445739061780, −6.04654415868436522178922617420, −4.88483635492894190675696365306, −3.22417332053968332142875178844, −1.92389867712298089128747555438, 0.923022372980503789475356786617, 2.79397349035486436924919882534, 3.83739045491954046624579769882, 4.64812659034720706308540859000, 6.44260162853193179711710261026, 7.07376755723031528527329842468, 8.588427710985871845803452479278, 9.335443119869730571501353499963, 9.991514284036754545431681400579, 10.69118280250641345641483885844

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