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Properties

Label 2-430-43.41-c1-0-6
Degree $2$
Conductor $430$
Sign $0.166 + 0.986i$
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  + (−0.900 + 0.433i)2-s + (0.371 + 0.179i)3-s + (0.623 − 0.781i)4-s + (0.222 − 0.974i)5-s − 0.412·6-s − 2.38·7-s + (−0.222 + 0.974i)8-s + (−1.76 − 2.21i)9-s + (0.222 + 0.974i)10-s + (1.90 + 2.38i)11-s + (0.371 − 0.179i)12-s + (1.49 − 6.53i)13-s + (2.14 − 1.03i)14-s + (0.257 − 0.322i)15-s + (−0.222 − 0.974i)16-s + (−1.01 − 4.45i)17-s + ⋯
L(s)  = 1  + (−0.637 + 0.306i)2-s + (0.214 + 0.103i)3-s + (0.311 − 0.390i)4-s + (0.0995 − 0.436i)5-s − 0.168·6-s − 0.901·7-s + (−0.0786 + 0.344i)8-s + (−0.588 − 0.737i)9-s + (0.0703 + 0.308i)10-s + (0.574 + 0.720i)11-s + (0.107 − 0.0516i)12-s + (0.413 − 1.81i)13-s + (0.574 − 0.276i)14-s + (0.0664 − 0.0832i)15-s + (−0.0556 − 0.243i)16-s + (−0.246 − 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.166 + 0.986i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.166 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.599159 - 0.506472i\)
\(L(\frac12)\) \(\approx\) \(0.599159 - 0.506472i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
43 \( 1 + (-3.21 + 5.71i)T \)
good3 \( 1 + (-0.371 - 0.179i)T + (1.87 + 2.34i)T^{2} \)
7 \( 1 + 2.38T + 7T^{2} \)
11 \( 1 + (-1.90 - 2.38i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.49 + 6.53i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + (1.01 + 4.45i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + (2.18 - 2.74i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (2.15 + 2.70i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-0.583 + 0.280i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (-2.69 + 1.29i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + 6.29T + 37T^{2} \)
41 \( 1 + (-7.21 + 3.47i)T + (25.5 - 32.0i)T^{2} \)
47 \( 1 + (-7.22 + 9.05i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.43 - 6.27i)T + (-47.7 + 22.9i)T^{2} \)
59 \( 1 + (-0.208 - 0.915i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (2.54 + 1.22i)T + (38.0 + 47.6i)T^{2} \)
67 \( 1 + (6.91 - 8.66i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-2.78 + 3.48i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (2.99 - 13.1i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 - 0.820T + 79T^{2} \)
83 \( 1 + (-6.12 - 2.94i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-8.98 - 4.32i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (5.31 + 6.66i)T + (-21.5 + 94.5i)T^{2} \)
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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

−10.61280299706515958421663245482, −9.919021478628955541056418277014, −9.097790061051788582116187971156, −8.422936776579264966598042830622, −7.31317266365510849179434894250, −6.28392439138132898468017350985, −5.48925258211636142373074218182, −3.89894150571314945700458722947, −2.64127104922786505963838112950, −0.58618439823486796927535698808, 1.84989440414472470020896243028, 3.09903739828373262557539734624, 4.21900908614683281871936272413, 6.12242606512771295179347320524, 6.62120228290513843864741071890, 7.84837091460080212541183713795, 8.888675031881390456505046502435, 9.339449669018473284246890917238, 10.58797512880149594890186974947, 11.16722037605882934267581087166

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