Properties

Label 2-930-465.464-c1-0-1
Degree $2$
Conductor $930$
Sign $-0.893 - 0.448i$
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  + 2-s + (−1.02 − 1.39i)3-s + 4-s + (−0.467 + 2.18i)5-s + (−1.02 − 1.39i)6-s − 1.43i·7-s + 8-s + (−0.884 + 2.86i)9-s + (−0.467 + 2.18i)10-s − 6.14·11-s + (−1.02 − 1.39i)12-s − 2.15·13-s − 1.43i·14-s + (3.52 − 1.59i)15-s + 16-s + 0.937i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.593 − 0.804i)3-s + 0.5·4-s + (−0.209 + 0.977i)5-s + (−0.419 − 0.568i)6-s − 0.540i·7-s + 0.353·8-s + (−0.294 + 0.955i)9-s + (−0.147 + 0.691i)10-s − 1.85·11-s + (−0.296 − 0.402i)12-s − 0.597·13-s − 0.382i·14-s + (0.910 − 0.412i)15-s + 0.250·16-s + 0.227i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.893 - 0.448i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.893 - 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0317621 + 0.134131i\)
\(L(\frac12)\) \(\approx\) \(0.0317621 + 0.134131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (1.02 + 1.39i)T \)
5 \( 1 + (0.467 - 2.18i)T \)
31 \( 1 + (3.50 + 4.32i)T \)
good7 \( 1 + 1.43iT - 7T^{2} \)
11 \( 1 + 6.14T + 11T^{2} \)
13 \( 1 + 2.15T + 13T^{2} \)
17 \( 1 - 0.937iT - 17T^{2} \)
19 \( 1 + 5.64T + 19T^{2} \)
23 \( 1 - 0.266iT - 23T^{2} \)
29 \( 1 - 3.72T + 29T^{2} \)
37 \( 1 + 7.85T + 37T^{2} \)
41 \( 1 - 3.31iT - 41T^{2} \)
43 \( 1 + 0.581T + 43T^{2} \)
47 \( 1 + 4.74T + 47T^{2} \)
53 \( 1 - 6.07iT - 53T^{2} \)
59 \( 1 - 5.39iT - 59T^{2} \)
61 \( 1 - 9.71iT - 61T^{2} \)
67 \( 1 + 9.90iT - 67T^{2} \)
71 \( 1 + 1.07iT - 71T^{2} \)
73 \( 1 - 3.14T + 73T^{2} \)
79 \( 1 - 2.07iT - 79T^{2} \)
83 \( 1 - 8.35iT - 83T^{2} \)
89 \( 1 + 6.18T + 89T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−10.60897109790134368329948545172, −10.15334034726145384377807588155, −8.306291977645947419816596643301, −7.58313366043090927248113459234, −7.01162444782596747097049381344, −6.14530062098899670176252881656, −5.29695931669508889059800388623, −4.30603952096522283744212130180, −2.91615188579880523249293122029, −2.12212534748322855762061682607, 0.04919889210427305436080867973, 2.26052950696685976914616198165, 3.49399317177685698622753329314, 4.72542820187403498722119196227, 5.09373880522074693874983490184, 5.81393583472239680757614214696, 6.98435309836840716269107569160, 8.176103978153140989244990017046, 8.829529149817730392407138051056, 9.947735830674716690002200322026

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