Properties

Label 2-430-43.21-c1-0-9
Degree $2$
Conductor $430$
Sign $0.485 + 0.874i$
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 + (2.29 − 1.10i)3-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s − 2.54·6-s + 3.65·7-s + (−0.222 − 0.974i)8-s + (2.17 − 2.72i)9-s + (−0.222 + 0.974i)10-s + (−0.380 + 0.477i)11-s + (2.29 + 1.10i)12-s + (0.400 + 1.75i)13-s + (−3.29 − 1.58i)14-s + (−1.58 − 1.99i)15-s + (−0.222 + 0.974i)16-s + (−0.257 + 1.12i)17-s + ⋯
L(s)  = 1  + (−0.637 − 0.306i)2-s + (1.32 − 0.637i)3-s + (0.311 + 0.390i)4-s + (−0.0995 − 0.436i)5-s − 1.03·6-s + 1.38·7-s + (−0.0786 − 0.344i)8-s + (0.723 − 0.907i)9-s + (−0.0703 + 0.308i)10-s + (−0.114 + 0.143i)11-s + (0.662 + 0.318i)12-s + (0.111 + 0.487i)13-s + (−0.881 − 0.424i)14-s + (−0.409 − 0.513i)15-s + (−0.0556 + 0.243i)16-s + (−0.0624 + 0.273i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.485 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.485 + 0.874i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.485 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49121 - 0.877576i\)
\(L(\frac12)\) \(\approx\) \(1.49121 - 0.877576i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.07 - 2.45i)T \)
good3 \( 1 + (-2.29 + 1.10i)T + (1.87 - 2.34i)T^{2} \)
7 \( 1 - 3.65T + 7T^{2} \)
11 \( 1 + (0.380 - 0.477i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-0.400 - 1.75i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (0.257 - 1.12i)T + (-15.3 - 7.37i)T^{2} \)
19 \( 1 + (1.64 + 2.06i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (-0.414 + 0.519i)T + (-5.11 - 22.4i)T^{2} \)
29 \( 1 + (5.23 + 2.52i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (-7.07 - 3.40i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 + (5.63 + 2.71i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (4.65 + 5.83i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.182 - 0.800i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 + (-0.0174 + 0.0765i)T + (-53.1 - 25.5i)T^{2} \)
61 \( 1 + (-10.2 + 4.95i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (1.85 + 2.32i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (-7.34 - 9.20i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.320 - 1.40i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 + (8.85 - 4.26i)T + (51.7 - 64.8i)T^{2} \)
89 \( 1 + (-10.5 + 5.09i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (6.72 - 8.43i)T + (-21.5 - 94.5i)T^{2} \)
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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

−11.04060333028501394180607480050, −9.921758415473571789581253231893, −8.739830672241786770868331634215, −8.508587954620363372963778808484, −7.67068242090055767766364683090, −6.80997909596640810634880580032, −5.04188005487677124623332494897, −3.80089364415039653002639722888, −2.33328107866675192338280156175, −1.49163142259358494014073352284, 1.87079108624734984886055240591, 3.11926141480367875357363954768, 4.36409279312872492418522747045, 5.54775470739613935441068270782, 7.05839079947203920118060385421, 8.080988615787725786388314167706, 8.368325491598135532241380922642, 9.373272198197400747289540193118, 10.27937718719218532101783837797, 10.97090044886257612589450513299

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