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Properties

Label 2-930-31.25-c1-0-0
Degree $2$
Conductor $930$
Sign $-0.879 + 0.475i$
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  − 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (−0.695 − 1.20i)7-s − 8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s + (−2.61 + 4.53i)11-s + (0.5 + 0.866i)12-s + (−0.390 + 0.676i)13-s + (0.695 + 1.20i)14-s − 0.999·15-s + 16-s + (−1.80 − 3.12i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.288 + 0.499i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (−0.262 − 0.455i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.158 − 0.273i)10-s + (−0.789 + 1.36i)11-s + (0.144 + 0.249i)12-s + (−0.108 + 0.187i)13-s + (0.185 + 0.321i)14-s − 0.258·15-s + 0.250·16-s + (−0.437 − 0.758i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.879 + 0.475i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.879 + 0.475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0198073 - 0.0782176i\)
\(L(\frac12)\) \(\approx\) \(0.0198073 - 0.0782176i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (5.54 - 0.527i)T \)
good7 \( 1 + (0.695 + 1.20i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.61 - 4.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.390 - 0.676i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.80 + 3.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.72 + 6.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.39T + 23T^{2} \)
29 \( 1 + 2.84T + 29T^{2} \)
37 \( 1 + (-2.80 - 4.85i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.50 - 9.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.92 + 6.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.17T + 47T^{2} \)
53 \( 1 + (-3.81 + 6.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.08 + 3.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 9.45T + 61T^{2} \)
67 \( 1 + (2.80 - 4.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.60 + 2.78i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.92 + 6.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.93 - 12.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.21T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−10.42192585536930964225105127895, −9.702610957220854957632268747197, −9.054781870513933615600127524660, −8.080935309689468668418033847239, −7.10469743170991911950960880693, −6.79356362402632121127436257253, −5.15107489570603082739643358572, −4.36305101590847907561901960653, −3.04990005647335337851843356199, −2.11664973311103435555120863408, 0.04299369751476599777834216751, 1.66261993017766728665428778203, 2.86528406495466777377553116107, 3.92009140314902500324853090463, 5.61649871989578060818825006330, 6.06422796288955784546991584416, 7.29693125398827872155148124977, 8.122579258695617440014675050229, 8.620622626344795642909902572157, 9.322448285814415160321790000663

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