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Properties

Label 2-966-7.4-c1-0-5
Degree $2$
Conductor $966$
Sign $-0.659 - 0.751i$
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 + (1.91 + 3.31i)5-s + 0.999·6-s + (−1.87 + 1.86i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.91 − 3.31i)10-s + (−2.63 + 4.57i)11-s + (−0.499 − 0.866i)12-s + 4.35·13-s + (2.55 + 0.690i)14-s − 3.82·15-s + (−0.5 − 0.866i)16-s + (1.23 − 2.13i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.856 + 1.48i)5-s + 0.408·6-s + (−0.708 + 0.705i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.605 − 1.04i)10-s + (−0.795 + 1.37i)11-s + (−0.144 − 0.249i)12-s + 1.20·13-s + (0.682 + 0.184i)14-s − 0.988·15-s + (−0.125 − 0.216i)16-s + (0.299 − 0.518i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.659 - 0.751i$
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: \((2,\ 966,\ (\ :1/2),\ -0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.412010 + 0.909233i\)
\(L(\frac12)\) \(\approx\) \(0.412010 + 0.909233i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.87 - 1.86i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1.91 - 3.31i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.63 - 4.57i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.35T + 13T^{2} \)
17 \( 1 + (-1.23 + 2.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.960 - 1.66i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 1.35T + 29T^{2} \)
31 \( 1 + (-0.718 + 1.24i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.31 + 5.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.92T + 41T^{2} \)
43 \( 1 + 8.21T + 43T^{2} \)
47 \( 1 + (-3.49 - 6.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.44 + 2.50i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.14 - 3.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.71 + 9.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.66 - 4.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.64T + 71T^{2} \)
73 \( 1 + (-0.218 + 0.378i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.679 + 1.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.37T + 83T^{2} \)
89 \( 1 + (-6.14 - 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.298T + 97T^{2} \)
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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

−10.31598852397630248388570072950, −9.706991692007031382521702133420, −9.117632461279518259090233242432, −7.81059501578399106629148711491, −6.84695279272988831685643352987, −6.06825794259664944381746615137, −5.19085337365639316869999150861, −3.73033545897871746575920757806, −2.86527656539661304553450314180, −2.00812086188858870684426302183, 0.55657478244018201094036547041, 1.48679863622441519156311285902, 3.35821958215022254496590548441, 4.72301341662018203096732242319, 5.78070269303079462716554091966, 6.01196739075732748747052914376, 7.14084390896114129046147429812, 8.400627876768756518096665988409, 8.518028311056392940869010903264, 9.601918407336123129231392082746

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