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Properties

Label 2-430-43.16-c1-0-2
Degree $2$
Conductor $430$
Sign $-0.707 + 0.706i$
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 + (−0.635 + 2.78i)3-s + (−0.900 + 0.433i)4-s + (0.623 + 0.781i)5-s − 2.85·6-s − 1.93·7-s + (−0.623 − 0.781i)8-s + (−4.63 − 2.23i)9-s + (−0.623 + 0.781i)10-s + (−1.80 − 0.870i)11-s + (−0.635 − 2.78i)12-s + (1.61 + 2.02i)13-s + (−0.430 − 1.88i)14-s + (−2.57 + 1.23i)15-s + (0.623 − 0.781i)16-s + (0.243 − 0.305i)17-s + ⋯
L(s)  = 1  + (0.157 + 0.689i)2-s + (−0.366 + 1.60i)3-s + (−0.450 + 0.216i)4-s + (0.278 + 0.349i)5-s − 1.16·6-s − 0.731·7-s + (−0.220 − 0.276i)8-s + (−1.54 − 0.744i)9-s + (−0.197 + 0.247i)10-s + (−0.544 − 0.262i)11-s + (−0.183 − 0.803i)12-s + (0.446 + 0.560i)13-s + (−0.115 − 0.504i)14-s + (−0.664 + 0.319i)15-s + (0.155 − 0.195i)16-s + (0.0591 − 0.0741i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.707 + 0.706i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.707 + 0.706i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327698 - 0.791851i\)
\(L(\frac12)\) \(\approx\) \(0.327698 - 0.791851i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.79 - 4.47i)T \)
good3 \( 1 + (0.635 - 2.78i)T + (-2.70 - 1.30i)T^{2} \)
7 \( 1 + 1.93T + 7T^{2} \)
11 \( 1 + (1.80 + 0.870i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.61 - 2.02i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.243 + 0.305i)T + (-3.78 - 16.5i)T^{2} \)
19 \( 1 + (3.37 - 1.62i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-3.78 - 1.82i)T + (14.3 + 17.9i)T^{2} \)
29 \( 1 + (-0.974 - 4.26i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (0.607 + 2.65i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 - 7.06T + 37T^{2} \)
41 \( 1 + (0.806 + 3.53i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (6.54 - 3.15i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-1.66 + 2.09i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (5.04 - 6.32i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (0.817 - 3.58i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (-13.8 + 6.69i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (12.8 - 6.19i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-4.36 - 5.47i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 + (3.46 - 15.1i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (2.52 - 11.0i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-1.68 - 0.809i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.37150719920297043615474625442, −10.71089769747352801093702462164, −9.803950659793103215113119682734, −9.243667580185122752651996277150, −8.225030596620880925157241813679, −6.77482444561539825750997212411, −5.95253454969071238373284310635, −5.05812460861012043877634793432, −4.02285615957670542705569920889, −3.07034755148447208930905691652, 0.53145522205712192078860494737, 1.95348344468144307812082531273, 3.08568713509247469985476304356, 4.84651482454902179768420817748, 5.99165781292892141421237936982, 6.67726833343951252625808677573, 7.84738800030003813240909567310, 8.678434803350349951920042243111, 9.845684110605313305144110540709, 10.80150304995881509056375532292

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