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Properties

Label 2-930-31.8-c1-0-0
Degree $2$
Conductor $930$
Sign $0.541 - 0.840i$
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  + (0.309 − 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s − 5-s − 0.999·6-s + (−3.72 − 2.70i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (0.659 + 0.479i)11-s + (−0.309 + 0.951i)12-s + (1.93 + 5.96i)13-s + (−3.72 + 2.70i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−4.62 + 3.36i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.178 − 0.549i)3-s + (−0.404 − 0.293i)4-s − 0.447·5-s − 0.408·6-s + (−1.40 − 1.02i)7-s + (−0.286 + 0.207i)8-s + (−0.269 + 0.195i)9-s + (−0.0977 + 0.300i)10-s + (0.198 + 0.144i)11-s + (−0.0892 + 0.274i)12-s + (0.537 + 1.65i)13-s + (−0.996 + 0.723i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (−1.12 + 0.815i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.541 - 0.840i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.541 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.297321 + 0.162199i\)
\(L(\frac12)\) \(\approx\) \(0.297321 + 0.162199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + T \)
31 \( 1 + (-4.44 + 3.35i)T \)
good7 \( 1 + (3.72 + 2.70i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.659 - 0.479i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.93 - 5.96i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.62 - 3.36i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.0338 + 0.104i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.09 + 2.24i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.434 + 1.33i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 6.85T + 37T^{2} \)
41 \( 1 + (-0.403 + 1.24i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (0.288 - 0.888i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-3.21 - 9.88i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.68 - 6.30i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.39 - 4.30i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 4.02T + 61T^{2} \)
67 \( 1 + 8.54T + 67T^{2} \)
71 \( 1 + (11.6 - 8.49i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (8.60 + 6.25i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-7.39 + 5.37i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.05 - 9.39i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-3.64 - 2.64i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.72 - 2.70i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.44922522045925569751361242494, −9.349748361172745682844068373182, −8.798176659180239635156247304407, −7.50413231797632017495658653250, −6.58845657045629284596974240159, −6.25576011179128882934896364208, −4.45135149692886303952602930088, −3.97641618099919568825547489210, −2.79577421141936915166877954651, −1.37418736817007008142737809737, 0.15809276048580369159308930280, 2.95516506296401547095241702400, 3.45537778082157512335237085381, 4.83439086206750038649471227654, 5.62321649463382989360289095949, 6.39695757030608690557498143555, 7.20971565151980644903597452640, 8.491260083971764805144563197922, 8.914088679983430768557754281414, 9.816142454205902996366848302866

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