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Properties

Label 2-966-7.4-c1-0-20
Degree $2$
Conductor $966$
Sign $0.832 + 0.553i$
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 + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.739 − 1.28i)5-s + 0.999·6-s + (1.32 − 2.29i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.739 − 1.28i)10-s + (−2.85 + 4.94i)11-s + (0.499 + 0.866i)12-s + 6.35·13-s + 2.64·14-s − 1.47·15-s + (−0.5 − 0.866i)16-s + (2.06 − 3.57i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.330 − 0.572i)5-s + 0.408·6-s + (0.499 − 0.866i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.233 − 0.405i)10-s + (−0.859 + 1.48i)11-s + (0.144 + 0.249i)12-s + 1.76·13-s + 0.707·14-s − 0.381·15-s + (−0.125 − 0.216i)16-s + (0.500 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89689 - 0.573267i\)
\(L(\frac12)\) \(\approx\) \(1.89689 - 0.573267i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-1.32 + 2.29i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.739 + 1.28i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.85 - 4.94i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.35T + 13T^{2} \)
17 \( 1 + (-2.06 + 3.57i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.69 + 2.93i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 2.39T + 29T^{2} \)
31 \( 1 + (-3.90 + 6.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.17 + 7.23i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.39T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (5.11 + 8.85i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.260 - 0.451i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.68 - 6.38i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.52 - 4.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.70 - 13.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.87T + 71T^{2} \)
73 \( 1 + (4.46 - 7.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.71 - 11.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.22T + 83T^{2} \)
89 \( 1 + (-3.21 - 5.57i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−9.855112718819374346270795845027, −8.761454938454153719590629468396, −8.119101448815859513767645120864, −7.40904237100922077282777691210, −6.75773268392422275303852070998, −5.58158364469713357710326598669, −4.57755167854189546532859823736, −3.95882541029332949109262696145, −2.45463746133463861279947110657, −0.883687605193698128324155961773, 1.52641132320620995268383451907, 3.13751036255344832119267435600, 3.37183496764510524353310868538, 4.70608806255920420578703078056, 5.78663639606028398761840815919, 6.25969164793082155657446743503, 8.160690664459739404833677797455, 8.289156908115399802426730225857, 9.244094297888275007354155466805, 10.56397991013698218953782110181

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