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Properties

Label 2-930-93.68-c1-0-3
Degree $2$
Conductor $930$
Sign $-0.923 - 0.383i$
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  − i·2-s + (−0.763 + 1.55i)3-s − 4-s + (−0.866 − 0.5i)5-s + (1.55 + 0.763i)6-s + (0.262 + 0.455i)7-s + i·8-s + (−1.83 − 2.37i)9-s + (−0.5 + 0.866i)10-s + (0.205 − 0.356i)11-s + (0.763 − 1.55i)12-s + (1.67 + 0.964i)13-s + (0.455 − 0.262i)14-s + (1.43 − 0.964i)15-s + 16-s + (1.92 + 3.32i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.440 + 0.897i)3-s − 0.5·4-s + (−0.387 − 0.223i)5-s + (0.634 + 0.311i)6-s + (0.0993 + 0.172i)7-s + 0.353i·8-s + (−0.611 − 0.791i)9-s + (−0.158 + 0.273i)10-s + (0.0620 − 0.107i)11-s + (0.220 − 0.448i)12-s + (0.463 + 0.267i)13-s + (0.121 − 0.0702i)14-s + (0.371 − 0.249i)15-s + 0.250·16-s + (0.465 + 0.806i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.923 - 0.383i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.923 - 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0251813 + 0.126178i\)
\(L(\frac12)\) \(\approx\) \(0.0251813 + 0.126178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.763 - 1.55i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (1.00 - 5.47i)T \)
good7 \( 1 + (-0.262 - 0.455i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.205 + 0.356i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.67 - 0.964i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.92 - 3.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.20 + 2.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.98T + 23T^{2} \)
29 \( 1 + 7.39T + 29T^{2} \)
37 \( 1 + (3.96 - 2.28i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (10.9 + 6.32i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.27 - 4.77i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.2iT - 47T^{2} \)
53 \( 1 + (-4.96 + 8.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (8.43 - 4.87i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 3.42iT - 61T^{2} \)
67 \( 1 + (2.97 - 5.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.72 - 3.30i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.0947 - 0.0547i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.7 - 7.92i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.89 - 5.02i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 16.9T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−10.27942045952646826976853243837, −9.982057075203460907885124247049, −8.625780479624284331787126522193, −8.534476290193400598849338524167, −6.96343760348870545045223064213, −5.80933738237755084672372635136, −5.07232236837012908857649681929, −3.95726645539627094992913961528, −3.46832262883987820799350845225, −1.78873171639791749112787907137, 0.06328812240111821171486222402, 1.73974612455691402310922618928, 3.34033516459564588941445637564, 4.52440816189806041895414920499, 5.67359517603352301860463894195, 6.23216409266196251339087569157, 7.28990849794035656083978692238, 7.76402230954994642368499044316, 8.505141933255356555922787535810, 9.651493317791250579610778106326

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