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Properties

Label 2-430-5.4-c1-0-5
Degree $2$
Conductor $430$
Sign $-0.404 - 0.914i$
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 + 3.26i·3-s − 4-s + (0.904 + 2.04i)5-s + 3.26·6-s + 1.22i·7-s + i·8-s − 7.63·9-s + (2.04 − 0.904i)10-s + 4.91·11-s − 3.26i·12-s − 3.77i·13-s + 1.22·14-s + (−6.66 + 2.95i)15-s + 16-s + 6.24i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.88i·3-s − 0.5·4-s + (0.404 + 0.914i)5-s + 1.33·6-s + 0.462i·7-s + 0.353i·8-s − 2.54·9-s + (0.646 − 0.286i)10-s + 1.48·11-s − 0.941i·12-s − 1.04i·13-s + 0.326·14-s + (−1.72 + 0.761i)15-s + 0.250·16-s + 1.51i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.404 - 0.914i$
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: \((2,\ 430,\ (\ :1/2),\ -0.404 - 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.703141 + 1.07995i\)
\(L(\frac12)\) \(\approx\) \(0.703141 + 1.07995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.904 - 2.04i)T \)
43 \( 1 - iT \)
good3 \( 1 - 3.26iT - 3T^{2} \)
7 \( 1 - 1.22iT - 7T^{2} \)
11 \( 1 - 4.91T + 11T^{2} \)
13 \( 1 + 3.77iT - 13T^{2} \)
17 \( 1 - 6.24iT - 17T^{2} \)
19 \( 1 + 5.03T + 19T^{2} \)
23 \( 1 + 4.72iT - 23T^{2} \)
29 \( 1 + 4.33T + 29T^{2} \)
31 \( 1 - 1.96T + 31T^{2} \)
37 \( 1 - 4.33iT - 37T^{2} \)
41 \( 1 - 6.72T + 41T^{2} \)
47 \( 1 + 4.48iT - 47T^{2} \)
53 \( 1 - 1.85iT - 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 9.35T + 61T^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + 1.14iT - 73T^{2} \)
79 \( 1 - 15.7T + 79T^{2} \)
83 \( 1 + 9.21iT - 83T^{2} \)
89 \( 1 - 3.91T + 89T^{2} \)
97 \( 1 - 5.00iT - 97T^{2} \)
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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

−11.07572533150720786754587493013, −10.47574808944537949851486905478, −9.964677183548046907254628365637, −9.001627036107090719812580782148, −8.374391662301794372637965852149, −6.34558267437983112540814073737, −5.60477679515118539343419242614, −4.22666467516374269657825076894, −3.60681921071243685020827900725, −2.44070232165272164359186378987, 0.861770821271889521128100405313, 2.02509391645407905580753883313, 4.10623330598031073691314450573, 5.48565450909164916200483240503, 6.46222467464096960170939850530, 7.02335485972315846010459690624, 7.88767991787231349095078375001, 8.983726587448561891767226700030, 9.351021167898266728098261454236, 11.32739442669559228408859257896

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