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Properties

Label 2-966-21.20-c1-0-41
Degree $2$
Conductor $966$
Sign $0.268 + 0.963i$
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  − i·2-s + (−0.952 + 1.44i)3-s − 4-s + 3.39·5-s + (1.44 + 0.952i)6-s + (1.73 − 1.99i)7-s + i·8-s + (−1.18 − 2.75i)9-s − 3.39i·10-s + 1.55i·11-s + (0.952 − 1.44i)12-s − 6.18i·13-s + (−1.99 − 1.73i)14-s + (−3.23 + 4.91i)15-s + 16-s − 2.89·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.549 + 0.835i)3-s − 0.5·4-s + 1.51·5-s + (0.590 + 0.388i)6-s + (0.656 − 0.753i)7-s + 0.353i·8-s + (−0.395 − 0.918i)9-s − 1.07i·10-s + 0.469i·11-s + (0.274 − 0.417i)12-s − 1.71i·13-s + (−0.533 − 0.464i)14-s + (−0.834 + 1.26i)15-s + 0.250·16-s − 0.701·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.268 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28471 - 0.975507i\)
\(L(\frac12)\) \(\approx\) \(1.28471 - 0.975507i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.952 - 1.44i)T \)
7 \( 1 + (-1.73 + 1.99i)T \)
23 \( 1 + iT \)
good5 \( 1 - 3.39T + 5T^{2} \)
11 \( 1 - 1.55iT - 11T^{2} \)
13 \( 1 + 6.18iT - 13T^{2} \)
17 \( 1 + 2.89T + 17T^{2} \)
19 \( 1 + 6.93iT - 19T^{2} \)
29 \( 1 - 1.06iT - 29T^{2} \)
31 \( 1 - 7.18iT - 31T^{2} \)
37 \( 1 + 6.46T + 37T^{2} \)
41 \( 1 + 4.43T + 41T^{2} \)
43 \( 1 - 7.07T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 - 2.78iT - 53T^{2} \)
59 \( 1 + 9.07T + 59T^{2} \)
61 \( 1 + 13.7iT - 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 - 5.72iT - 71T^{2} \)
73 \( 1 + 11.2iT - 73T^{2} \)
79 \( 1 - 0.738T + 79T^{2} \)
83 \( 1 - 2.20T + 83T^{2} \)
89 \( 1 - 3.01T + 89T^{2} \)
97 \( 1 - 6.59iT - 97T^{2} \)
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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

−10.10652376033510974553296658786, −9.278981324651312323135661083636, −8.607287399334628839207123649002, −7.21215277323227154748731875270, −6.17928232415407857363664984061, −5.09167373035664173734025934583, −4.83787476201636024858685529291, −3.42140431016953748421505724381, −2.29715340131776326328506838872, −0.843313898445362466011629981836, 1.60306316490195525079531374325, 2.26855187111489684555228129519, 4.31709262600170444561335542512, 5.47847095295610772820120305372, 5.91105749660037673155726614500, 6.56359919876802820322882936410, 7.51675255444182939657826906423, 8.568577670643204328719134861109, 9.125155125033474898510252089476, 10.06629505765939438359339934240

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