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Properties

Label 2-966-161.68-c1-0-18
Degree $2$
Conductor $966$
Sign $0.619 + 0.785i$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
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  + (0.5 − 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (−1.36 + 2.35i)5-s − 0.999i·6-s + (2.49 − 0.871i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (1.36 + 2.35i)10-s + (2.53 − 1.46i)11-s + (−0.866 − 0.499i)12-s − 1.35i·13-s + (0.494 − 2.59i)14-s + 2.72i·15-s + (−0.5 + 0.866i)16-s + (3.08 + 5.35i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.608 + 1.05i)5-s − 0.408i·6-s + (0.944 − 0.329i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.430 + 0.745i)10-s + (0.763 − 0.440i)11-s + (−0.249 − 0.144i)12-s − 0.376i·13-s + (0.132 − 0.694i)14-s + 0.702i·15-s + (−0.125 + 0.216i)16-s + (0.749 + 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.619 + 0.785i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.619 + 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09306 - 1.01509i\)
\(L(\frac12)\) \(\approx\) \(2.09306 - 1.01509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-2.49 + 0.871i)T \)
23 \( 1 + (-4.29 + 2.12i)T \)
good5 \( 1 + (1.36 - 2.35i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.53 + 1.46i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.35iT - 13T^{2} \)
17 \( 1 + (-3.08 - 5.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.541 - 0.938i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 7.21T + 29T^{2} \)
31 \( 1 + (-8.29 + 4.79i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.810 - 0.467i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.65iT - 41T^{2} \)
43 \( 1 + 9.65iT - 43T^{2} \)
47 \( 1 + (-6.68 - 3.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.10 - 1.79i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.816 + 0.471i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.62 + 4.54i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.27 - 3.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.73T + 71T^{2} \)
73 \( 1 + (8.70 - 5.02i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.01 + 4.62i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.05T + 83T^{2} \)
89 \( 1 + (7.90 - 13.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.72T + 97T^{2} \)
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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

−10.22170641530196431762531665776, −9.023398792860927080588780368548, −8.165775797549861498868668424739, −7.48539238075689351538759351398, −6.53541711456776191338198360394, −5.53558641832155230294839078238, −4.11382722531634895879392320605, −3.60047062865593515935992731034, −2.49259169560937531880080526686, −1.20391933011395509430825150067, 1.31491399778511494204145975770, 2.96567458418949001715426110962, 4.27703564059058182553539122992, 4.73312406420063021885458717876, 5.54606680578859450425166527621, 6.95970332698386042261644821202, 7.68425836209558015701022730190, 8.465346727722747498190810773615, 9.084383838934522699588015541231, 9.744115152318700379389885066567

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