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Properties

Label 2-966-161.68-c1-0-7
Degree $2$
Conductor $966$
Sign $-0.944 - 0.328i$
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 + (−0.286 + 0.496i)5-s − 0.999i·6-s + (1.61 + 2.09i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−0.286 − 0.496i)10-s + (−1.12 + 0.652i)11-s + (0.866 + 0.499i)12-s + 1.60i·13-s + (−2.62 + 0.347i)14-s − 0.573i·15-s + (−0.5 + 0.866i)16-s + (0.403 + 0.699i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.128 + 0.222i)5-s − 0.408i·6-s + (0.609 + 0.792i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.0907 − 0.157i)10-s + (−0.340 + 0.196i)11-s + (0.249 + 0.144i)12-s + 0.444i·13-s + (−0.700 + 0.0928i)14-s − 0.148i·15-s + (−0.125 + 0.216i)16-s + (0.0979 + 0.169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.944 - 0.328i$
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: \((2,\ 966,\ (\ :1/2),\ -0.944 - 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.139437 + 0.824224i\)
\(L(\frac12)\) \(\approx\) \(0.139437 + 0.824224i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 2.09i)T \)
23 \( 1 + (-3.11 - 3.64i)T \)
good5 \( 1 + (0.286 - 0.496i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.12 - 0.652i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.60iT - 13T^{2} \)
17 \( 1 + (-0.403 - 0.699i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.97 + 3.41i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 5.64T + 29T^{2} \)
31 \( 1 + (2.57 - 1.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.07 - 2.35i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.23iT - 41T^{2} \)
43 \( 1 + 1.28iT - 43T^{2} \)
47 \( 1 + (6.14 + 3.54i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.74 - 1.58i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.9 - 6.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.19 + 2.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (12.1 - 6.99i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 + (-3.49 + 2.01i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.39 - 3.68i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.99T + 83T^{2} \)
89 \( 1 + (-1.46 + 2.53i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.12T + 97T^{2} \)
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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

−10.34605762360552322330078809734, −9.315078003697834298103322858253, −8.911082035831811558145008778906, −7.72501750094207519743069965201, −7.12703560615680763125987539978, −6.04584484430244533907442511352, −5.27220616601110722147957860645, −4.57210589577726429934048624847, −3.11366413759678719917203620704, −1.56170914988455931413129745549, 0.49223038142409859082377827898, 1.68709207271159792326996211029, 3.12063126814182942877532360943, 4.29933373993257748726975755688, 5.12203134851542859095561058678, 6.20970622368262679606136389916, 7.44281205620483914039220619816, 7.86879718775761179085785747272, 8.825434643628402349936398144259, 9.851739717390329367272604803283

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