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Properties

Label 2-430-43.10-c1-0-9
Degree $2$
Conductor $430$
Sign $0.345 + 0.938i$
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 + (−0.387 − 0.359i)3-s + (−0.900 + 0.433i)4-s + (−0.988 + 0.149i)5-s + (−0.264 + 0.457i)6-s + (1.56 + 2.71i)7-s + (0.623 + 0.781i)8-s + (−0.203 − 2.71i)9-s + (0.365 + 0.930i)10-s + (2.80 + 1.35i)11-s + (0.505 + 0.155i)12-s + (1.80 − 4.60i)13-s + (2.29 − 2.12i)14-s + (0.436 + 0.297i)15-s + (0.623 − 0.781i)16-s + (1.40 + 0.212i)17-s + ⋯
L(s)  = 1  + (−0.157 − 0.689i)2-s + (−0.223 − 0.207i)3-s + (−0.450 + 0.216i)4-s + (−0.442 + 0.0666i)5-s + (−0.107 + 0.186i)6-s + (0.591 + 1.02i)7-s + (0.220 + 0.276i)8-s + (−0.0677 − 0.904i)9-s + (0.115 + 0.294i)10-s + (0.846 + 0.407i)11-s + (0.145 + 0.0449i)12-s + (0.501 − 1.27i)13-s + (0.613 − 0.569i)14-s + (0.112 + 0.0768i)15-s + (0.155 − 0.195i)16-s + (0.341 + 0.0515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.345 + 0.938i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.345 + 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.965356 - 0.673155i\)
\(L(\frac12)\) \(\approx\) \(0.965356 - 0.673155i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.90 - 4.34i)T \)
good3 \( 1 + (0.387 + 0.359i)T + (0.224 + 2.99i)T^{2} \)
7 \( 1 + (-1.56 - 2.71i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.80 - 1.35i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.80 + 4.60i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (-1.40 - 0.212i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (-0.0862 + 1.15i)T + (-18.7 - 2.83i)T^{2} \)
23 \( 1 + (-5.44 + 3.71i)T + (8.40 - 21.4i)T^{2} \)
29 \( 1 + (-3.77 + 3.50i)T + (2.16 - 28.9i)T^{2} \)
31 \( 1 + (4.98 + 1.53i)T + (25.6 + 17.4i)T^{2} \)
37 \( 1 + (3.66 - 6.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.974 + 4.26i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (-8.11 + 3.90i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-2.87 - 7.33i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (-8.09 + 10.1i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (7.55 - 2.33i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (1.00 - 13.4i)T + (-66.2 - 9.98i)T^{2} \)
71 \( 1 + (3.14 + 2.14i)T + (25.9 + 66.0i)T^{2} \)
73 \( 1 + (0.259 - 0.661i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (2.20 + 3.82i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.15 + 2.92i)T + (6.20 + 82.7i)T^{2} \)
89 \( 1 + (1.50 + 1.39i)T + (6.65 + 88.7i)T^{2} \)
97 \( 1 + (-16.3 - 7.85i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.15790337361996855454632064941, −10.21979639374563736421962744076, −9.034929298800384852825607971724, −8.567580340813028541071434304427, −7.38910598150774052034674926346, −6.18677414135350477197062379802, −5.14535320275487608342117847651, −3.85197661756577203196919554655, −2.71108351241866199493327898257, −1.03143021455172485675256245680, 1.36378152266494608690197096580, 3.75014619032390762489416084793, 4.55585721015590586142078127939, 5.60332014743454153861102294348, 6.92324399614801857749319514442, 7.48664137626006400679576067743, 8.562685676988529569080770594836, 9.307455471070351445130093799832, 10.65094492868844336864859149400, 11.08607195731566920840497985495

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