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Properties

Label 2-930-31.8-c1-0-6
Degree $2$
Conductor $930$
Sign $-0.784 - 0.620i$
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 + (0.170 + 0.123i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (3.26 + 2.37i)11-s + (0.309 − 0.951i)12-s + (1.56 + 4.80i)13-s + (−0.170 + 0.123i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (4.78 − 3.47i)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 + (0.0642 + 0.0467i)7-s + (0.286 − 0.207i)8-s + (−0.269 + 0.195i)9-s + (0.0977 − 0.300i)10-s + (0.985 + 0.716i)11-s + (0.0892 − 0.274i)12-s + (0.432 + 1.33i)13-s + (−0.0454 + 0.0330i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (1.16 − 0.843i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.784 - 0.620i$
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: \((2,\ 930,\ (\ :1/2),\ -0.784 - 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408089 + 1.17441i\)
\(L(\frac12)\) \(\approx\) \(0.408089 + 1.17441i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.91 + 4.74i)T \)
good7 \( 1 + (-0.170 - 0.123i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-3.26 - 2.37i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.56 - 4.80i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-4.78 + 3.47i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.30 + 4.01i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (5.69 - 4.13i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.77 - 8.54i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 3.63T + 37T^{2} \)
41 \( 1 + (1.63 - 5.03i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (2.89 - 8.91i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-3.40 - 10.4i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.184 + 0.133i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.47 - 7.62i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 9.93T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + (1.67 - 1.21i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.95 + 2.87i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-8.41 + 6.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.946 - 2.91i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (6.89 + 5.01i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-7.71 - 5.60i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.05887250578946935941727174541, −9.348261261161595949239367500144, −8.948812044333959040617253705957, −7.74956534417579663968649880208, −7.14004344999212883318413693529, −6.20619830847738665580647875993, −5.07244941528427958924038869078, −4.27345871311669662381948543761, −3.36380274933329695222200908568, −1.55839751716899089825485065804, 0.67018704066761935837509059283, 1.90158343235274390724055643910, 3.46519508533359956718911110254, 3.78651408619551780551182118430, 5.49306589370206489347218978157, 6.21581317746517389218047351325, 7.50485386820729536147793978415, 8.230109830349351562131250175494, 8.670242360573678240679658999099, 9.953301951997218033904520140335

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