Properties

Label 2-930-93.11-c1-0-25
Degree $2$
Conductor $930$
Sign $0.996 + 0.0893i$
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  + (0.951 − 0.309i)2-s + (1.43 − 0.973i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (1.06 − 1.36i)6-s + (0.396 + 3.76i)7-s + (0.587 − 0.809i)8-s + (1.10 − 2.78i)9-s + (−0.669 + 0.743i)10-s + (3.50 + 1.55i)11-s + (0.586 − 1.62i)12-s + (1.20 + 5.67i)13-s + (1.54 + 3.46i)14-s + (−0.753 + 1.55i)15-s + (0.309 − 0.951i)16-s + (−2.94 + 1.31i)17-s + ⋯
L(s)  = 1  + (0.672 − 0.218i)2-s + (0.826 − 0.562i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (0.433 − 0.558i)6-s + (0.149 + 1.42i)7-s + (0.207 − 0.286i)8-s + (0.367 − 0.929i)9-s + (−0.211 + 0.235i)10-s + (1.05 + 0.470i)11-s + (0.169 − 0.470i)12-s + (0.334 + 1.57i)13-s + (0.412 + 0.925i)14-s + (−0.194 + 0.402i)15-s + (0.0772 − 0.237i)16-s + (−0.714 + 0.318i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.996 + 0.0893i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.996 + 0.0893i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.12713 - 0.139923i\)
\(L(\frac12)\) \(\approx\) \(3.12713 - 0.139923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (-1.43 + 0.973i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-0.212 + 5.56i)T \)
good7 \( 1 + (-0.396 - 3.76i)T + (-6.84 + 1.45i)T^{2} \)
11 \( 1 + (-3.50 - 1.55i)T + (7.36 + 8.17i)T^{2} \)
13 \( 1 + (-1.20 - 5.67i)T + (-11.8 + 5.28i)T^{2} \)
17 \( 1 + (2.94 - 1.31i)T + (11.3 - 12.6i)T^{2} \)
19 \( 1 + (-6.45 - 1.37i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (5.27 + 3.82i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.71 + 8.34i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 + (2.61 + 1.51i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.97 - 4.48i)T + (4.28 + 40.7i)T^{2} \)
43 \( 1 + (-1.30 + 6.13i)T + (-39.2 - 17.4i)T^{2} \)
47 \( 1 + (10.3 + 3.36i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.652 - 6.21i)T + (-51.8 - 11.0i)T^{2} \)
59 \( 1 + (-3.01 + 2.71i)T + (6.16 - 58.6i)T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 + (2.41 + 4.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.247 + 0.0260i)T + (69.4 + 14.7i)T^{2} \)
73 \( 1 + (-2.26 + 5.08i)T + (-48.8 - 54.2i)T^{2} \)
79 \( 1 + (-0.661 - 1.48i)T + (-52.8 + 58.7i)T^{2} \)
83 \( 1 + (1.71 - 1.90i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-0.635 + 0.461i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (1.23 - 0.900i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.754693826724729091614936482206, −9.239727355453118174294819120793, −8.427606021205960770256447528039, −7.46292654734666745005602967187, −6.49750565243741744267561748202, −5.95352687770743818262055371595, −4.39969375449623703375193950477, −3.75941076114133897215823559368, −2.42801762213787211584801727370, −1.77118273945933490179988123136, 1.28767788062460460046156297187, 3.29550107209071353249819138371, 3.57322602390915377808399538417, 4.62530262735304862810177650217, 5.45958511762164497339134681808, 6.85495537198531049115409404517, 7.57587125790530233216240540163, 8.232652013384723293511084286142, 9.212105868597059549091475676029, 10.12072747858102479510667219199

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