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Properties

Label 2-930-155.149-c1-0-0
Degree $2$
Conductor $930$
Sign $-0.0736 - 0.997i$
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 + (−0.866 + 0.5i)3-s − 4-s + (0.0369 − 2.23i)5-s + (0.5 + 0.866i)6-s + (0.230 − 0.133i)7-s + i·8-s + (0.499 − 0.866i)9-s + (−2.23 − 0.0369i)10-s + (−2.16 + 3.74i)11-s + (0.866 − 0.5i)12-s + (−3.19 − 1.84i)13-s + (−0.133 − 0.230i)14-s + (1.08 + 1.95i)15-s + 16-s + (−3.14 + 1.81i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.499 + 0.288i)3-s − 0.5·4-s + (0.0165 − 0.999i)5-s + (0.204 + 0.353i)6-s + (0.0872 − 0.0503i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.707 − 0.0116i)10-s + (−0.652 + 1.13i)11-s + (0.249 − 0.144i)12-s + (−0.885 − 0.511i)13-s + (−0.0356 − 0.0616i)14-s + (0.280 + 0.504i)15-s + 0.250·16-s + (−0.762 + 0.440i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0736 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0736 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0736 - 0.997i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.0736 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.172098 + 0.185272i\)
\(L(\frac12)\) \(\approx\) \(0.172098 + 0.185272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.0369 + 2.23i)T \)
31 \( 1 + (4.12 - 3.73i)T \)
good7 \( 1 + (-0.230 + 0.133i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.16 - 3.74i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.19 + 1.84i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.14 - 1.81i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.627 - 1.08i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2.23iT - 23T^{2} \)
29 \( 1 - 8.58T + 29T^{2} \)
37 \( 1 + (5.89 - 3.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.09 + 5.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.46 - 4.88i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.03iT - 47T^{2} \)
53 \( 1 + (-6.70 - 3.87i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.855 - 1.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 5.59T + 61T^{2} \)
67 \( 1 + (-1.14 - 0.658i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.22 - 9.04i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.38 + 3.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.29 - 3.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-9.52 - 5.49i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + 11.1iT - 97T^{2} \)
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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

−10.24808267980947011712537113579, −9.711500894912497638599549245401, −8.765843409943729404829108797713, −7.965796015533605755285729540123, −6.90717017940369156427516953235, −5.60138525347539166963067349595, −4.78913219655365356484126557907, −4.32663028183221910373638437387, −2.77703687272125620621211866089, −1.50271021113957873856732754221, 0.12515662483056110063251645696, 2.28740060695181425113378395291, 3.47187755882953367960776407276, 4.82772040376599746058329928574, 5.61716120464316760981288004557, 6.58595254837249653823196153376, 7.09320004922619526490538056613, 7.966343100964446502181499030086, 8.870175125669717160069777850840, 9.926981168662607308160106148512

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