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Properties

Label 2-966-161.68-c1-0-25
Degree $2$
Conductor $966$
Sign $0.640 + 0.767i$
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 + (2.07 − 3.58i)5-s + 0.999i·6-s + (1.78 − 1.95i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (2.07 + 3.58i)10-s + (2.94 − 1.70i)11-s + (−0.866 − 0.499i)12-s − 0.0118i·13-s + (0.795 + 2.52i)14-s − 4.14i·15-s + (−0.5 + 0.866i)16-s + (3.74 + 6.48i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.926 − 1.60i)5-s + 0.408i·6-s + (0.675 − 0.737i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.654 + 1.13i)10-s + (0.889 − 0.513i)11-s + (−0.249 − 0.144i)12-s − 0.00329i·13-s + (0.212 + 0.674i)14-s − 1.06i·15-s + (−0.125 + 0.216i)16-s + (0.908 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.640 + 0.767i$
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: \((2,\ 966,\ (\ :1/2),\ 0.640 + 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83870 - 0.860228i\)
\(L(\frac12)\) \(\approx\) \(1.83870 - 0.860228i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 + 1.95i)T \)
23 \( 1 + (4.79 + 0.0326i)T \)
good5 \( 1 + (-2.07 + 3.58i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.94 + 1.70i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.0118iT - 13T^{2} \)
17 \( 1 + (-3.74 - 6.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.10 - 3.64i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 3.05T + 29T^{2} \)
31 \( 1 + (-5.66 + 3.27i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.217 + 0.125i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.42iT - 41T^{2} \)
43 \( 1 - 6.14iT - 43T^{2} \)
47 \( 1 + (-1.66 - 0.963i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.02 - 3.48i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.24 - 2.44i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.73 + 3.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.67 + 3.27i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 + (12.4 - 7.20i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.9 - 8.03i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.21T + 83T^{2} \)
89 \( 1 + (-2.47 + 4.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.85T + 97T^{2} \)
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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

−9.768008681500017564429280474122, −8.859519548232000907503629556965, −8.188049536730274430221894112604, −7.82191288410861115251759160748, −6.21954054265427730849996644140, −5.90301434830440839674434652656, −4.61549859766349926080868910720, −3.89768620205958152112877007300, −1.71706987428041252296891353287, −1.16455560585040710242809036758, 1.85900617739018357392865846722, 2.57305607952734773390003868567, 3.43762695020057986787286375263, 4.75318874313268423729291371416, 5.86150830387737958069178907190, 6.93194160953652883106429190096, 7.59024550551732836813202927735, 8.804877897083806778391192894548, 9.452349911968387295496312238587, 10.07765748781070513022203414780

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