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Properties

Label 2-430-215.42-c1-0-7
Degree $2$
Conductor $430$
Sign $0.999 + 0.00996i$
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  + (−0.707 + 0.707i)2-s + (−0.325 − 0.325i)3-s − 1.00i·4-s + (2.06 − 0.863i)5-s + 0.459·6-s + (−2.30 + 2.30i)7-s + (0.707 + 0.707i)8-s − 2.78i·9-s + (−0.847 + 2.06i)10-s + 5.14·11-s + (−0.325 + 0.325i)12-s + (−1.01 + 1.01i)13-s − 3.26i·14-s + (−0.951 − 0.389i)15-s − 1.00·16-s + (−1.10 − 1.10i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.187 − 0.187i)3-s − 0.500i·4-s + (0.922 − 0.386i)5-s + 0.187·6-s + (−0.872 + 0.872i)7-s + (0.250 + 0.250i)8-s − 0.929i·9-s + (−0.268 + 0.654i)10-s + 1.55·11-s + (−0.0938 + 0.0938i)12-s + (−0.282 + 0.282i)13-s − 0.872i·14-s + (−0.245 − 0.100i)15-s − 0.250·16-s + (−0.266 − 0.266i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.999 + 0.00996i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.999 + 0.00996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16601 - 0.00580900i\)
\(L(\frac12)\) \(\approx\) \(1.16601 - 0.00580900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-2.06 + 0.863i)T \)
43 \( 1 + (1.13 - 6.45i)T \)
good3 \( 1 + (0.325 + 0.325i)T + 3iT^{2} \)
7 \( 1 + (2.30 - 2.30i)T - 7iT^{2} \)
11 \( 1 - 5.14T + 11T^{2} \)
13 \( 1 + (1.01 - 1.01i)T - 13iT^{2} \)
17 \( 1 + (1.10 + 1.10i)T + 17iT^{2} \)
19 \( 1 - 3.19T + 19T^{2} \)
23 \( 1 + (-5.12 + 5.12i)T - 23iT^{2} \)
29 \( 1 + 1.35T + 29T^{2} \)
31 \( 1 - 2.94T + 31T^{2} \)
37 \( 1 + (-7.77 + 7.77i)T - 37iT^{2} \)
41 \( 1 + 0.959T + 41T^{2} \)
47 \( 1 + (-9.36 - 9.36i)T + 47iT^{2} \)
53 \( 1 + (3.85 - 3.85i)T - 53iT^{2} \)
59 \( 1 - 7.97iT - 59T^{2} \)
61 \( 1 - 1.09iT - 61T^{2} \)
67 \( 1 + (8.56 + 8.56i)T + 67iT^{2} \)
71 \( 1 + 13.6iT - 71T^{2} \)
73 \( 1 + (5.49 + 5.49i)T + 73iT^{2} \)
79 \( 1 + 3.25iT - 79T^{2} \)
83 \( 1 + (8.65 - 8.65i)T - 83iT^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 + (5.34 + 5.34i)T + 97iT^{2} \)
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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

−11.15132526464925150296782674458, −9.774214698164906012163363183758, −9.134937571245079507755889722695, −8.987712813488830728969616015768, −7.22567706492034598134132436737, −6.26326267038838754759815997584, −5.98914749181698422872712557161, −4.50806943483051190364024642221, −2.79519727988317948929033072101, −1.11440056063804038712612073824, 1.37395177504436894522673440967, 2.91349822725597967707957423337, 4.04688969620393636982291301167, 5.45951943537340333970377296963, 6.68868911887557167549554476593, 7.31063870432704854261315829623, 8.722434662333347826522698488782, 9.756933302753089228239914974518, 10.02845951153710067801969552835, 11.04414728714202070586654676004

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