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Properties

Label 2-966-161.32-c1-0-10
Degree $2$
Conductor $966$
Sign $-0.768 - 0.639i$
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  + (0.723 + 0.690i)2-s + (0.981 − 0.189i)3-s + (0.0475 + 0.998i)4-s + (−0.487 + 0.383i)5-s + (0.841 + 0.540i)6-s + (−2.45 + 0.987i)7-s + (−0.654 + 0.755i)8-s + (0.928 − 0.371i)9-s + (−0.616 − 0.0589i)10-s + (−2.95 + 2.81i)11-s + (0.235 + 0.971i)12-s + (2.24 + 4.92i)13-s + (−2.45 − 0.978i)14-s + (−0.405 + 0.468i)15-s + (−0.995 + 0.0950i)16-s + (0.348 − 0.179i)17-s + ⋯
L(s)  = 1  + (0.511 + 0.487i)2-s + (0.566 − 0.109i)3-s + (0.0237 + 0.499i)4-s + (−0.217 + 0.171i)5-s + (0.343 + 0.220i)6-s + (−0.927 + 0.373i)7-s + (−0.231 + 0.267i)8-s + (0.309 − 0.123i)9-s + (−0.195 − 0.0186i)10-s + (−0.891 + 0.849i)11-s + (0.0680 + 0.280i)12-s + (0.623 + 1.36i)13-s + (−0.656 − 0.261i)14-s + (−0.104 + 0.120i)15-s + (−0.248 + 0.0237i)16-s + (0.0844 − 0.0435i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.768 - 0.639i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.768 - 0.639i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557110 + 1.53953i\)
\(L(\frac12)\) \(\approx\) \(0.557110 + 1.53953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.723 - 0.690i)T \)
3 \( 1 + (-0.981 + 0.189i)T \)
7 \( 1 + (2.45 - 0.987i)T \)
23 \( 1 + (-0.865 + 4.71i)T \)
good5 \( 1 + (0.487 - 0.383i)T + (1.17 - 4.85i)T^{2} \)
11 \( 1 + (2.95 - 2.81i)T + (0.523 - 10.9i)T^{2} \)
13 \( 1 + (-2.24 - 4.92i)T + (-8.51 + 9.82i)T^{2} \)
17 \( 1 + (-0.348 + 0.179i)T + (9.86 - 13.8i)T^{2} \)
19 \( 1 + (4.63 + 2.39i)T + (11.0 + 15.4i)T^{2} \)
29 \( 1 + (-2.23 - 1.43i)T + (12.0 + 26.3i)T^{2} \)
31 \( 1 + (0.168 + 0.486i)T + (-24.3 + 19.1i)T^{2} \)
37 \( 1 + (7.21 - 2.88i)T + (26.7 - 25.5i)T^{2} \)
41 \( 1 + (-1.67 - 11.6i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (-2.26 - 2.60i)T + (-6.11 + 42.5i)T^{2} \)
47 \( 1 + (-0.662 - 1.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.801 - 1.12i)T + (-17.3 + 50.0i)T^{2} \)
59 \( 1 + (-5.77 - 0.551i)T + (57.9 + 11.1i)T^{2} \)
61 \( 1 + (-0.991 - 0.191i)T + (56.6 + 22.6i)T^{2} \)
67 \( 1 + (-1.87 + 7.74i)T + (-59.5 - 30.7i)T^{2} \)
71 \( 1 + (-8.24 + 2.41i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (-0.0885 - 1.85i)T + (-72.6 + 6.93i)T^{2} \)
79 \( 1 + (-4.86 + 6.82i)T + (-25.8 - 74.6i)T^{2} \)
83 \( 1 + (-0.685 + 4.76i)T + (-79.6 - 23.3i)T^{2} \)
89 \( 1 + (-0.118 + 0.342i)T + (-69.9 - 55.0i)T^{2} \)
97 \( 1 + (-1.80 - 12.5i)T + (-93.0 + 27.3i)T^{2} \)
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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

−10.24355170883185299082138802021, −9.295567314320172221841429867477, −8.670804594371906409593291496004, −7.73075784117594095304978218720, −6.75610294487169413790613558203, −6.39459165017508256420973228173, −4.99991087217516533079461716087, −4.15349384567538732646137626337, −3.09644366209099290964114405473, −2.13235107859662109199619810900, 0.56808174725741276268010241874, 2.38813429612189707670665002247, 3.41455884782911223148284446909, 3.92108737683923408614400052151, 5.36214244032993013049119365012, 6.01131859818388122936965049930, 7.18858081410126793211945143843, 8.203571135771954310432504414330, 8.777964270652567782043790427235, 10.02877361365444220322102408140

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