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Properties

Label 2-930-31.4-c1-0-3
Degree $2$
Conductor $930$
Sign $0.944 - 0.329i$
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 + (−0.0930 + 0.0675i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (−4.78 + 3.47i)11-s + (0.309 + 0.951i)12-s + (−0.0677 + 0.208i)13-s + (0.0930 + 0.0675i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (6.20 + 4.50i)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 + (−0.0351 + 0.0255i)7-s + (0.286 + 0.207i)8-s + (−0.269 − 0.195i)9-s + (0.0977 + 0.300i)10-s + (−1.44 + 1.04i)11-s + (0.0892 + 0.274i)12-s + (−0.0187 + 0.0578i)13-s + (0.0248 + 0.0180i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (1.50 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.944 - 0.329i$
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: \((2,\ 930,\ (\ :1/2),\ 0.944 - 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928321 + 0.157377i\)
\(L(\frac12)\) \(\approx\) \(0.928321 + 0.157377i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.30 + 3.53i)T \)
good7 \( 1 + (0.0930 - 0.0675i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (4.78 - 3.47i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.0677 - 0.208i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-6.20 - 4.50i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.223 + 0.686i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.30 - 2.40i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.40 - 4.32i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + 11.2T + 37T^{2} \)
41 \( 1 + (-1.05 - 3.25i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-0.732 - 2.25i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (0.348 - 1.07i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.59 - 5.51i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.911 - 2.80i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 8.33T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + (-4.55 - 3.31i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-7.02 + 5.10i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.118 - 0.0862i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-2.78 - 8.58i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-10.9 + 7.95i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.42 - 1.76i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.27772370829304222616538653131, −9.364944527814754179067581034253, −8.301971106089091146009225263509, −7.75609053313060953891186101653, −7.01850976697733348381565060289, −5.65004724876012305019154992319, −4.73990647148167942053600314948, −3.50152370880779029085035568350, −2.57719907961154536352758954199, −1.32687436625787635886375345746, 0.50860081584230384522968096457, 2.79191882009134753108194447035, 3.66584543628389198884138101190, 5.10515038401850473093093109731, 5.39397346573032224442611738425, 6.71126372069068662860483892732, 7.68870585645593762800813940756, 8.274533581028601782197924852728, 8.996283341258993584038666078081, 10.15242880824345526742818832677

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