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Properties

Label 2-966-161.68-c1-0-14
Degree $2$
Conductor $966$
Sign $0.365 - 0.930i$
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 + (−0.856 + 1.48i)5-s + 0.999i·6-s + (0.871 + 2.49i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−0.856 − 1.48i)10-s + (4.71 − 2.72i)11-s + (−0.866 − 0.499i)12-s − 1.89i·13-s + (−2.59 − 0.494i)14-s + 1.71i·15-s + (−0.5 + 0.866i)16-s + (0.914 + 1.58i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.382 + 0.663i)5-s + 0.408i·6-s + (0.329 + 0.944i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.270 − 0.468i)10-s + (1.42 − 0.820i)11-s + (−0.249 − 0.144i)12-s − 0.525i·13-s + (−0.694 − 0.132i)14-s + 0.442i·15-s + (−0.125 + 0.216i)16-s + (0.221 + 0.384i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.365 - 0.930i$
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: \((2,\ 966,\ (\ :1/2),\ 0.365 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32340 + 0.902504i\)
\(L(\frac12)\) \(\approx\) \(1.32340 + 0.902504i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.871 - 2.49i)T \)
23 \( 1 + (-2.63 + 4.00i)T \)
good5 \( 1 + (0.856 - 1.48i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.71 + 2.72i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.89iT - 13T^{2} \)
17 \( 1 + (-0.914 - 1.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.07 - 1.85i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 + (0.763 - 0.440i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.21 - 5.32i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.9iT - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + (11.4 + 6.62i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.86 + 2.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-11.6 + 6.71i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.09 - 3.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.17 + 1.25i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.36T + 71T^{2} \)
73 \( 1 + (-9.32 + 5.38i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.82 - 5.67i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.05T + 83T^{2} \)
89 \( 1 + (-3.76 + 6.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.00T + 97T^{2} \)
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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

−9.878128989097767074698805883384, −9.190567154872935729529939097719, −8.257396325262270364631890093927, −7.983492045327561649766742032957, −6.59415012969560295496227028440, −6.32470113909576399061778724478, −5.10047992821311804767449065566, −3.78221812493471861049201936981, −2.82095079772818258926451791156, −1.31944160323318584020466988394, 0.966358784882030408448684819791, 2.12444723016408541492006539331, 3.78226498633562510006795694607, 4.14806744812270264921060305846, 5.11279280665372508427874397941, 6.83834115905638766121122394468, 7.43918649250918106758260099192, 8.396627954137292870893845721363, 9.271205743424907338832574431918, 9.583674258034205866033122901834

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