Properties

Label 2-966-23.12-c1-0-9
Degree $2$
Conductor $966$
Sign $0.880 + 0.474i$
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  + (−0.841 + 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (0.279 − 0.0819i)5-s + (0.415 + 0.909i)6-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (−0.190 + 0.219i)10-s + (1.72 + 1.10i)11-s + (−0.841 − 0.540i)12-s + (2.81 − 3.24i)13-s + (−0.959 − 0.281i)14-s + (−0.0414 − 0.288i)15-s + (−0.654 − 0.755i)16-s + (−0.928 − 2.03i)17-s + ⋯
L(s)  = 1  + (−0.594 + 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (0.124 − 0.0366i)5-s + (0.169 + 0.371i)6-s + (0.247 + 0.285i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.0602 + 0.0695i)10-s + (0.520 + 0.334i)11-s + (−0.242 − 0.156i)12-s + (0.779 − 0.900i)13-s + (−0.256 − 0.0752i)14-s + (−0.0106 − 0.0743i)15-s + (−0.163 − 0.188i)16-s + (−0.225 − 0.493i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.880 + 0.474i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.880 + 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28873 - 0.325181i\)
\(L(\frac12)\) \(\approx\) \(1.28873 - 0.325181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.841 - 0.540i)T \)
3 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (-2.48 - 4.09i)T \)
good5 \( 1 + (-0.279 + 0.0819i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (-1.72 - 1.10i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-2.81 + 3.24i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (0.928 + 2.03i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-0.997 + 2.18i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-1.60 - 3.52i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (0.775 + 5.39i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-9.76 - 2.86i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-8.76 + 2.57i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-0.756 + 5.26i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 + 3.78T + 47T^{2} \)
53 \( 1 + (5.29 + 6.10i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (5.68 - 6.56i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (1.45 + 10.1i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-13.2 + 8.54i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (-10.1 + 6.49i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-4.26 + 9.33i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (0.338 - 0.390i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-5.61 - 1.64i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-0.273 + 1.90i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (15.9 - 4.67i)T + (81.6 - 52.4i)T^{2} \)
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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

−9.525963984275333579381191892642, −9.254139327923607910704108943663, −8.097247793844320618489804994268, −7.61658029607277569688385016885, −6.62357145494709783667484193722, −5.85088390933317800744962730058, −4.94878455631989263244235353348, −3.45564528111463099345124088332, −2.15603516152482335177696536359, −0.915776674066667046530883090342, 1.21329735053027337723905380878, 2.56903527319362423744660581003, 3.85440520853365034685368936336, 4.44547900546166976768628649990, 5.94759945037974701531304543187, 6.64663508087105493826589757924, 7.896655285887447238648695179375, 8.535237328311531069497525843910, 9.368968498008747309230900237861, 9.965023783869496039421812029968

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