Properties

Label 2-930-31.14-c1-0-17
Degree $2$
Conductor $930$
Sign $-0.795 + 0.605i$
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 + (0.318 − 3.02i)7-s + (0.809 − 0.587i)8-s + (−0.104 − 0.994i)9-s + (−0.669 − 0.743i)10-s + (−3.16 + 1.41i)11-s + (−0.978 + 0.207i)12-s + (−3.78 − 0.804i)13-s + (2.77 + 1.23i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.972 − 0.433i)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 + (0.120 − 1.14i)7-s + (0.286 − 0.207i)8-s + (−0.0348 − 0.331i)9-s + (−0.211 − 0.235i)10-s + (−0.955 + 0.425i)11-s + (−0.282 + 0.0600i)12-s + (−1.04 − 0.223i)13-s + (0.742 + 0.330i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.235 − 0.105i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.795 + 0.605i$
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: \((2,\ 930,\ (\ :1/2),\ -0.795 + 0.605i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0862367 - 0.255517i\)
\(L(\frac12)\) \(\approx\) \(0.0862367 - 0.255517i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.36 + 1.48i)T \)
good7 \( 1 + (-0.318 + 3.02i)T + (-6.84 - 1.45i)T^{2} \)
11 \( 1 + (3.16 - 1.41i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (3.78 + 0.804i)T + (11.8 + 5.28i)T^{2} \)
17 \( 1 + (0.972 + 0.433i)T + (11.3 + 12.6i)T^{2} \)
19 \( 1 + (2.29 - 0.488i)T + (17.3 - 7.72i)T^{2} \)
23 \( 1 + (1.36 - 0.989i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.30 - 7.09i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + (1.30 + 2.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0605 + 0.0672i)T + (-4.28 + 40.7i)T^{2} \)
43 \( 1 + (4.05 - 0.862i)T + (39.2 - 17.4i)T^{2} \)
47 \( 1 + (0.499 + 1.53i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.0310 + 0.295i)T + (-51.8 + 11.0i)T^{2} \)
59 \( 1 + (-0.957 + 1.06i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 - 1.22T + 61T^{2} \)
67 \( 1 + (-7.47 + 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.44 + 13.7i)T + (-69.4 + 14.7i)T^{2} \)
73 \( 1 + (11.6 - 5.17i)T + (48.8 - 54.2i)T^{2} \)
79 \( 1 + (4.61 + 2.05i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (-2.71 - 3.01i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 + (12.1 + 8.82i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-6.45 - 4.68i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.749811000098854643569341319774, −8.713190326907789786600114056621, −7.67912809956770818430585043207, −7.40456360882919209060345999243, −6.66877570792448724692086888236, −5.40185059460843176736654733083, −4.46689068743575522990086378912, −3.33889444094472951440880665718, −1.96070892687470429992660681327, −0.11971637001640281560677048688, 2.09673296273316611756928129822, 2.79893433007854775907115725096, 4.07791166258578654732928634777, 5.01524779711060102019702279306, 5.78346321881477733589884713246, 7.30111042961083015696641372565, 8.295421033182807609892457898219, 8.717347380485981999251654901784, 9.616915575740866345020820704971, 10.25403358662353702710483293321

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