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Properties

Label 2-930-31.8-c1-0-1
Degree $2$
Conductor $930$
Sign $-0.860 + 0.508i$
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 + (1.31 + 0.957i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (−0.438 − 0.318i)11-s + (0.309 − 0.951i)12-s + (−1.26 − 3.90i)13-s + (−1.31 + 0.957i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−4.63 + 3.37i)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 + (0.498 + 0.361i)7-s + (0.286 − 0.207i)8-s + (−0.269 + 0.195i)9-s + (0.0977 − 0.300i)10-s + (−0.132 − 0.0960i)11-s + (0.0892 − 0.274i)12-s + (−0.352 − 1.08i)13-s + (−0.352 + 0.255i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (−1.12 + 0.817i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.860 + 0.508i$
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: \((2,\ 930,\ (\ :1/2),\ -0.860 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134246 - 0.490850i\)
\(L(\frac12)\) \(\approx\) \(0.134246 - 0.490850i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.47 - 1.00i)T \)
good7 \( 1 + (-1.31 - 0.957i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (0.438 + 0.318i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.26 + 3.90i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.63 - 3.37i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.96 - 6.04i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (6.13 - 4.46i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.0513 - 0.157i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 3.76T + 37T^{2} \)
41 \( 1 + (-2.66 + 8.21i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-0.369 + 1.13i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-0.679 - 2.09i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (11.3 - 8.26i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.83 + 5.64i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 7.49T + 67T^{2} \)
71 \( 1 + (8.09 - 5.88i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (5.10 + 3.70i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.55 + 3.30i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.17 - 9.77i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.10 - 1.52i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-15.1 - 10.9i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.47778871725895003369843975589, −9.647426851714502996876768813960, −8.673532082871124342458663519488, −8.083995282351769155003253034751, −7.46301203565521038947280715318, −6.08537945610770421992366826058, −5.48991748095802511020755297498, −4.38667013490751620307219444142, −3.55820118849861113518657743394, −1.97687841928968996172198760551, 0.23914802263668298805643916396, 1.87975261618902981261494191972, 2.76384205216231523444698089496, 4.26939428525260469587758897904, 4.71413965115890191557416807810, 6.37668833887165024661354781173, 7.17365717508945231091073623470, 7.945433732419868738118233018235, 8.849399878652639280398244061310, 9.447696556571305367698617458390

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