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Properties

Label 2-430-5.4-c1-0-16
Degree $2$
Conductor $430$
Sign $-0.116 + 0.993i$
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  − i·2-s + 2.99i·3-s − 4-s + (0.260 − 2.22i)5-s + 2.99·6-s − 4.62i·7-s + i·8-s − 5.96·9-s + (−2.22 − 0.260i)10-s − 3.48·11-s − 2.99i·12-s − 4.77i·13-s − 4.62·14-s + (6.64 + 0.780i)15-s + 16-s − 2.65i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.72i·3-s − 0.5·4-s + (0.116 − 0.993i)5-s + 1.22·6-s − 1.74i·7-s + 0.353i·8-s − 1.98·9-s + (−0.702 − 0.0824i)10-s − 1.05·11-s − 0.864i·12-s − 1.32i·13-s − 1.23·14-s + (1.71 + 0.201i)15-s + 0.250·16-s − 0.642i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.116 + 0.993i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.116 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.661660 - 0.743932i\)
\(L(\frac12)\) \(\approx\) \(0.661660 - 0.743932i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (-0.260 + 2.22i)T \)
43 \( 1 - iT \)
good3 \( 1 - 2.99iT - 3T^{2} \)
7 \( 1 + 4.62iT - 7T^{2} \)
11 \( 1 + 3.48T + 11T^{2} \)
13 \( 1 + 4.77iT - 13T^{2} \)
17 \( 1 + 2.65iT - 17T^{2} \)
19 \( 1 - 2.10T + 19T^{2} \)
23 \( 1 - 4.96iT - 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 + 3.61T + 31T^{2} \)
37 \( 1 + 5.36iT - 37T^{2} \)
41 \( 1 + 0.557T + 41T^{2} \)
47 \( 1 + 3.17iT - 47T^{2} \)
53 \( 1 + 2.74iT - 53T^{2} \)
59 \( 1 - 7.03T + 59T^{2} \)
61 \( 1 + 9.73T + 61T^{2} \)
67 \( 1 - 7.69iT - 67T^{2} \)
71 \( 1 - 6.01T + 71T^{2} \)
73 \( 1 - 8.26iT - 73T^{2} \)
79 \( 1 - 7.97T + 79T^{2} \)
83 \( 1 - 3.32iT - 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−10.61657608391000970690640276522, −10.13397434448762424333229692755, −9.563114077061500735883752091502, −8.435023649743690670822635210871, −7.57911464634078308833253130886, −5.45355238346582104068566480628, −4.90307466817295753179087694874, −3.98681392879146289833343208300, −3.08845846941106557875897665057, −0.62774919223114875605400322547, 2.04110062994970860626681617976, 2.86626495373158825504632749073, 5.12265903129234992238937264709, 6.34817298020369639353962243792, 6.43278619440779594540035199540, 7.66011121499857602021506326538, 8.348131829522919871635623462275, 9.205963246591723461125390826670, 10.60400102630114816528866687870, 11.82668761105086479154510373358

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