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Properties

Label 2-430-43.38-c1-0-6
Degree $2$
Conductor $430$
Sign $0.998 - 0.0619i$
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 + (1.95 + 0.601i)3-s + (−0.900 − 0.433i)4-s + (0.365 + 0.930i)5-s + (1.02 − 1.76i)6-s + (2.34 + 4.05i)7-s + (−0.623 + 0.781i)8-s + (0.965 + 0.658i)9-s + (0.988 − 0.149i)10-s + (1.10 − 0.530i)11-s + (−1.49 − 1.38i)12-s + (−3.31 − 0.499i)13-s + (4.47 − 1.38i)14-s + (0.152 + 2.03i)15-s + (0.623 + 0.781i)16-s + (−2.11 + 5.38i)17-s + ⋯
L(s)  = 1  + (0.157 − 0.689i)2-s + (1.12 + 0.347i)3-s + (−0.450 − 0.216i)4-s + (0.163 + 0.416i)5-s + (0.416 − 0.721i)6-s + (0.885 + 1.53i)7-s + (−0.220 + 0.276i)8-s + (0.321 + 0.219i)9-s + (0.312 − 0.0471i)10-s + (0.331 − 0.159i)11-s + (−0.432 − 0.400i)12-s + (−0.919 − 0.138i)13-s + (1.19 − 0.369i)14-s + (0.0393 + 0.525i)15-s + (0.155 + 0.195i)16-s + (−0.513 + 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.998 - 0.0619i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.998 - 0.0619i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15571 + 0.0667976i\)
\(L(\frac12)\) \(\approx\) \(2.15571 + 0.0667976i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.365 - 0.930i)T \)
43 \( 1 + (5.66 + 3.30i)T \)
good3 \( 1 + (-1.95 - 0.601i)T + (2.47 + 1.68i)T^{2} \)
7 \( 1 + (-2.34 - 4.05i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.10 + 0.530i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (3.31 + 0.499i)T + (12.4 + 3.83i)T^{2} \)
17 \( 1 + (2.11 - 5.38i)T + (-12.4 - 11.5i)T^{2} \)
19 \( 1 + (-2.58 + 1.76i)T + (6.94 - 17.6i)T^{2} \)
23 \( 1 + (-0.558 + 7.44i)T + (-22.7 - 3.42i)T^{2} \)
29 \( 1 + (-8.01 + 2.47i)T + (23.9 - 16.3i)T^{2} \)
31 \( 1 + (3.78 + 3.51i)T + (2.31 + 30.9i)T^{2} \)
37 \( 1 + (0.0399 - 0.0692i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.47 + 10.8i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-9.95 - 4.79i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (7.37 - 1.11i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (4.79 + 6.01i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-4.47 + 4.15i)T + (4.55 - 60.8i)T^{2} \)
67 \( 1 + (0.293 - 0.200i)T + (24.4 - 62.3i)T^{2} \)
71 \( 1 + (-0.716 - 9.56i)T + (-70.2 + 10.5i)T^{2} \)
73 \( 1 + (-1.28 - 0.193i)T + (69.7 + 21.5i)T^{2} \)
79 \( 1 + (-2.04 - 3.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.34 - 1.33i)T + (68.5 + 46.7i)T^{2} \)
89 \( 1 + (5.37 + 1.65i)T + (73.5 + 50.1i)T^{2} \)
97 \( 1 + (10.4 - 5.02i)T + (60.4 - 75.8i)T^{2} \)
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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

−11.16010360370185314906922662813, −10.24776140497455097034713412112, −9.219219125380937543473429273657, −8.691648644677056444367085565261, −7.940467315058771410064100532381, −6.31836336535899159205876287022, −5.18266421876743490082423739618, −4.05795383695230970578749423550, −2.69331003656872664223282976583, −2.18754053200515556955594135567, 1.41888766933557875419005394285, 3.13305195168061639780302701996, 4.41693394955332762957523150375, 5.18675660471813912592967859938, 6.95802574525003007663156860593, 7.46814821533339281244744848731, 8.157608523769678737146370560410, 9.195952273460471943709858165869, 9.896730313096285591498126862350, 11.20046445335014755910501611025

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