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Properties

Label 2-966-161.68-c1-0-6
Degree $2$
Conductor $966$
Sign $0.569 - 0.822i$
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 + (0.139 − 0.240i)5-s + 0.999i·6-s + (−2.42 − 1.05i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−0.139 − 0.240i)10-s + (−1.25 + 0.726i)11-s + (0.866 + 0.499i)12-s + 5.02i·13-s + (−2.12 + 1.57i)14-s + 0.278i·15-s + (−0.5 + 0.866i)16-s + (2.48 + 4.31i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.0622 − 0.107i)5-s + 0.408i·6-s + (−0.917 − 0.397i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.0439 − 0.0762i)10-s + (−0.379 + 0.219i)11-s + (0.249 + 0.144i)12-s + 1.39i·13-s + (−0.567 + 0.421i)14-s + 0.0718i·15-s + (−0.125 + 0.216i)16-s + (0.603 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.569 - 0.822i$
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: \((2,\ 966,\ (\ :1/2),\ 0.569 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.835787 + 0.437881i\)
\(L(\frac12)\) \(\approx\) \(0.835787 + 0.437881i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.42 + 1.05i)T \)
23 \( 1 + (4.78 + 0.335i)T \)
good5 \( 1 + (-0.139 + 0.240i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.25 - 0.726i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.02iT - 13T^{2} \)
17 \( 1 + (-2.48 - 4.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.02 + 1.78i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 5.24T + 29T^{2} \)
31 \( 1 + (-0.516 + 0.298i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.31 + 1.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.01iT - 41T^{2} \)
43 \( 1 - 6.84iT - 43T^{2} \)
47 \( 1 + (-8.05 - 4.65i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.65 - 4.99i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.40 + 1.38i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.995 - 1.72i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.81 - 3.35i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.995T + 71T^{2} \)
73 \( 1 + (13.8 - 7.97i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.24 + 1.29i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.53T + 83T^{2} \)
89 \( 1 + (3.83 - 6.64i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.93T + 97T^{2} \)
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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

−10.19723621325444794701681136383, −9.595355971621279033841713810003, −8.794167741765537270888442572926, −7.49049610021149046562533168804, −6.49817488751364079168530781328, −5.83971195146277204745269361603, −4.65027009334892298119933512419, −3.96818514537281057878340242558, −2.87667215300554671375165248650, −1.38789950604464387177430056917, 0.43484369850088041853515001686, 2.63482503595632805721871037522, 3.50434780062577824517458973064, 4.94748484177180567832408746920, 5.68622773426666848968218684576, 6.31252527569017314900115705225, 7.27499919868404424732065674567, 8.013336398767090351455809045933, 8.913081539108158146750950385798, 10.10325676813968016623834316359

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