Properties

Label 2-930-31.14-c1-0-0
Degree $2$
Conductor $930$
Sign $-0.223 + 0.974i$
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 + (−0.669 + 0.743i)3-s + (−0.809 − 0.587i)4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)6-s + (−0.220 + 2.10i)7-s + (0.809 − 0.587i)8-s + (−0.104 − 0.994i)9-s + (−0.669 − 0.743i)10-s + (1.49 − 0.664i)11-s + (0.978 − 0.207i)12-s + (−3.23 − 0.688i)13-s + (−1.93 − 0.859i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (0.273 + 0.121i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.386 + 0.429i)3-s + (−0.404 − 0.293i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (−0.0835 + 0.794i)7-s + (0.286 − 0.207i)8-s + (−0.0348 − 0.331i)9-s + (−0.211 − 0.235i)10-s + (0.449 − 0.200i)11-s + (0.282 − 0.0600i)12-s + (−0.897 − 0.190i)13-s + (−0.516 − 0.229i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.0663 + 0.0295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.223 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.223 + 0.974i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.223 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0730712 - 0.0917248i\)
\(L(\frac12)\) \(\approx\) \(0.0730712 - 0.0917248i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-5.07 + 2.28i)T \)
good7 \( 1 + (0.220 - 2.10i)T + (-6.84 - 1.45i)T^{2} \)
11 \( 1 + (-1.49 + 0.664i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (3.23 + 0.688i)T + (11.8 + 5.28i)T^{2} \)
17 \( 1 + (-0.273 - 0.121i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (5.71 - 1.21i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (7.09 - 5.15i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-1.04 + 3.22i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (-1.67 - 2.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.41 + 7.12i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (-2.22 + 0.473i)T + (39.2 - 17.4i)T^{2} \)
47 \( 1 + (3.64 + 11.2i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.744 + 7.08i)T + (-51.8 + 11.0i)T^{2} \)
59 \( 1 + (-1.07 + 1.19i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + 4.00T + 61T^{2} \)
67 \( 1 + (6.30 - 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.776 + 7.38i)T + (-69.4 + 14.7i)T^{2} \)
73 \( 1 + (5.59 - 2.49i)T + (48.8 - 54.2i)T^{2} \)
79 \( 1 + (7.74 + 3.44i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (1.43 + 1.59i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 + (13.5 + 9.84i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.88 - 2.82i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.24934008221148307268135262362, −9.990738423606798612964426852924, −8.889010352205469324073971828472, −8.210136157075220896971275887875, −7.23583436485834384494081354383, −6.25510582299125559777759859493, −5.70249484696450397357073687401, −4.61399267658384561970954063802, −3.64925211662688284037120880155, −2.17562870726851285180659379266, 0.06246563681307606430503781358, 1.46513736486171615992609533814, 2.71411844325793250443193769851, 4.24093220698533399826167241645, 4.63522209877637121198690545350, 6.13270814297396684493658355184, 6.96131308501496978427436684573, 7.88246294303735725642085440603, 8.626748906097459438949638284664, 9.662730030743836577166090310119

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