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Properties

Label 2-430-43.35-c1-0-10
Degree $2$
Conductor $430$
Sign $-0.430 - 0.902i$
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 + (−0.273 − 1.19i)3-s + (−0.900 − 0.433i)4-s + (−0.623 + 0.781i)5-s − 1.22·6-s − 4.39·7-s + (−0.623 + 0.781i)8-s + (1.34 − 0.647i)9-s + (0.623 + 0.781i)10-s + (−2.14 + 1.03i)11-s + (−0.273 + 1.19i)12-s + (−2.19 + 2.75i)13-s + (−0.978 + 4.28i)14-s + (1.10 + 0.532i)15-s + (0.623 + 0.781i)16-s + (0.329 + 0.412i)17-s + ⋯
L(s)  = 1  + (0.157 − 0.689i)2-s + (−0.157 − 0.690i)3-s + (−0.450 − 0.216i)4-s + (−0.278 + 0.349i)5-s − 0.501·6-s − 1.66·7-s + (−0.220 + 0.276i)8-s + (0.448 − 0.215i)9-s + (0.197 + 0.247i)10-s + (−0.647 + 0.311i)11-s + (−0.0788 + 0.345i)12-s + (−0.609 + 0.764i)13-s + (−0.261 + 1.14i)14-s + (0.285 + 0.137i)15-s + (0.155 + 0.195i)16-s + (0.0798 + 0.100i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.430 - 0.902i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.430 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0464987 + 0.0737097i\)
\(L(\frac12)\) \(\approx\) \(0.0464987 + 0.0737097i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.01 - 5.82i)T \)
good3 \( 1 + (0.273 + 1.19i)T + (-2.70 + 1.30i)T^{2} \)
7 \( 1 + 4.39T + 7T^{2} \)
11 \( 1 + (2.14 - 1.03i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.19 - 2.75i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-0.329 - 0.412i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + (2.22 + 1.07i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (2.56 - 1.23i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (0.718 - 3.14i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-0.986 + 4.32i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + 5.82T + 37T^{2} \)
41 \( 1 + (-1.69 + 7.43i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (7.70 + 3.70i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-0.376 - 0.471i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-6.16 - 7.72i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (1.65 + 7.23i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (7.12 + 3.43i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (13.8 + 6.68i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (3.11 - 3.90i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 2.64T + 79T^{2} \)
83 \( 1 + (1.79 + 7.85i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (0.629 + 2.75i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-9.72 + 4.68i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.43756185914393212493839977252, −9.862565301824893916228287605368, −8.997193524611705113016451280082, −7.52732487726669206349600309577, −6.81139675571369800310971240542, −5.93577480875514600359229074213, −4.38421828597154459153193367418, −3.29554169375489844035028422981, −2.10022856630931937744972796731, −0.05037569872370564566191189818, 3.00555447781153030234686936663, 4.06028543971350798459988443555, 5.12814172743146395443951689787, 6.05190854514205870671918711196, 7.09878268247596635465982272048, 8.048499404078703249012721541885, 9.115401893125582379195935325134, 10.04861700399176694642543317386, 10.42451240426483734135295046102, 11.95353966633823092750383047820

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