Properties

Label 2-930-93.92-c1-0-12
Degree $2$
Conductor $930$
Sign $-0.968 - 0.250i$
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 + (1.02 + 1.39i)3-s − 4-s + i·5-s + (−1.39 + 1.02i)6-s + 0.608·7-s i·8-s + (−0.901 + 2.86i)9-s − 10-s + 1.47·11-s + (−1.02 − 1.39i)12-s + 4.49i·13-s + 0.608i·14-s + (−1.39 + 1.02i)15-s + 16-s + 0.826·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.591 + 0.806i)3-s − 0.5·4-s + 0.447i·5-s + (−0.570 + 0.418i)6-s + 0.230·7-s − 0.353i·8-s + (−0.300 + 0.953i)9-s − 0.316·10-s + 0.444·11-s + (−0.295 − 0.403i)12-s + 1.24i·13-s + 0.162i·14-s + (−0.360 + 0.264i)15-s + 0.250·16-s + 0.200·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212864 + 1.67462i\)
\(L(\frac12)\) \(\approx\) \(0.212864 + 1.67462i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.02 - 1.39i)T \)
5 \( 1 - iT \)
31 \( 1 + (4.31 - 3.52i)T \)
good7 \( 1 - 0.608T + 7T^{2} \)
11 \( 1 - 1.47T + 11T^{2} \)
13 \( 1 - 4.49iT - 13T^{2} \)
17 \( 1 - 0.826T + 17T^{2} \)
19 \( 1 + 0.806T + 19T^{2} \)
23 \( 1 + 1.31T + 23T^{2} \)
29 \( 1 - 1.72T + 29T^{2} \)
37 \( 1 + 2.79iT - 37T^{2} \)
41 \( 1 + 7.02iT - 41T^{2} \)
43 \( 1 - 12.7iT - 43T^{2} \)
47 \( 1 + 4.11iT - 47T^{2} \)
53 \( 1 - 7.69T + 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 - 7.71iT - 61T^{2} \)
67 \( 1 - 4.39T + 67T^{2} \)
71 \( 1 + 4.02iT - 71T^{2} \)
73 \( 1 + 4.96iT - 73T^{2} \)
79 \( 1 - 4.39iT - 79T^{2} \)
83 \( 1 - 2.04T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 - 2.89T + 97T^{2} \)
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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

−10.25933909571759025184251496302, −9.414516105823463472331261802124, −8.854461678844916143387989342126, −7.977069061923572426967285981872, −7.10218553307946829514589657051, −6.24019414232359941473160452232, −5.12068795351892737586959023703, −4.24881425629289723174561564468, −3.42606063588665814118137108053, −2.00127677683135219353384667960, 0.75875257530681558189913229083, 1.93051901291419506064590175078, 3.06083181138082071283534439547, 4.00700914554264760247216418917, 5.26302753636646960519087983039, 6.19771121066504997654130790127, 7.39562041552873823256427131929, 8.151239123238264944150675959688, 8.792372951857835925943939632250, 9.646460232128085053997633767500

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