Properties

Label 2-930-31.4-c1-0-23
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: $\chi_{930} (841, \cdot )$
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

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

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