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Properties

Label 2-930-15.2-c1-0-16
Degree $2$
Conductor $930$
Sign $0.781 - 0.624i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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 + (0.0792 − 1.73i)3-s − 1.00i·4-s + (−1.29 + 1.82i)5-s + (1.16 + 1.27i)6-s + (2.17 + 2.17i)7-s + (0.707 + 0.707i)8-s + (−2.98 − 0.274i)9-s + (−0.370 − 2.20i)10-s − 4.45i·11-s + (−1.73 − 0.0792i)12-s + (0.974 − 0.974i)13-s − 3.07·14-s + (3.04 + 2.38i)15-s − 1.00·16-s + (−0.387 + 0.387i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.0457 − 0.998i)3-s − 0.500i·4-s + (−0.580 + 0.814i)5-s + (0.476 + 0.522i)6-s + (0.822 + 0.822i)7-s + (0.250 + 0.250i)8-s + (−0.995 − 0.0913i)9-s + (−0.117 − 0.697i)10-s − 1.34i·11-s + (−0.499 − 0.0228i)12-s + (0.270 − 0.270i)13-s − 0.822·14-s + (0.787 + 0.616i)15-s − 0.250·16-s + (−0.0940 + 0.0940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.781 - 0.624i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.781 - 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06726 + 0.374136i\)
\(L(\frac12)\) \(\approx\) \(1.06726 + 0.374136i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.0792 + 1.73i)T \)
5 \( 1 + (1.29 - 1.82i)T \)
31 \( 1 + T \)
good7 \( 1 + (-2.17 - 2.17i)T + 7iT^{2} \)
11 \( 1 + 4.45iT - 11T^{2} \)
13 \( 1 + (-0.974 + 0.974i)T - 13iT^{2} \)
17 \( 1 + (0.387 - 0.387i)T - 17iT^{2} \)
19 \( 1 - 7.08iT - 19T^{2} \)
23 \( 1 + (-4.52 - 4.52i)T + 23iT^{2} \)
29 \( 1 - 6.82T + 29T^{2} \)
37 \( 1 + (2.48 + 2.48i)T + 37iT^{2} \)
41 \( 1 - 9.64iT - 41T^{2} \)
43 \( 1 + (1.87 - 1.87i)T - 43iT^{2} \)
47 \( 1 + (-9.31 + 9.31i)T - 47iT^{2} \)
53 \( 1 + (-10.1 - 10.1i)T + 53iT^{2} \)
59 \( 1 - 4.06T + 59T^{2} \)
61 \( 1 - 7.35T + 61T^{2} \)
67 \( 1 + (5.24 + 5.24i)T + 67iT^{2} \)
71 \( 1 - 12.5iT - 71T^{2} \)
73 \( 1 + (-7.93 + 7.93i)T - 73iT^{2} \)
79 \( 1 - 17.0iT - 79T^{2} \)
83 \( 1 + (2.57 + 2.57i)T + 83iT^{2} \)
89 \( 1 - 2.48T + 89T^{2} \)
97 \( 1 + (-10.2 - 10.2i)T + 97iT^{2} \)
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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

−10.25251169331929819862698572517, −8.831390943203092626491147716395, −8.327626392114516949385058819474, −7.78068436724487316202258228366, −6.86005132971480573227720109181, −5.97054547564531088461930290299, −5.39995203641056846446510538378, −3.61008116570349241788268934006, −2.53036227886923364158920415092, −1.14012800283552277791457778266, 0.77979444610505453211744600178, 2.39290110882422335655173186010, 3.82925060932294136478849801970, 4.64157401500754333732652089348, 4.96739906871624633467517554676, 6.87849310060806454753721813608, 7.62279967741429811752174205249, 8.757599981253322245460575541418, 8.955500726169267447613784410116, 10.09789926693592431215381783880

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