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Properties

Label 2-966-161.68-c1-0-3
Degree $2$
Conductor $966$
Sign $-0.0878 - 0.996i$
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 + (−1.28 + 2.22i)5-s − 0.999i·6-s + (−1.74 − 1.98i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (1.28 + 2.22i)10-s + (−3.74 + 2.16i)11-s + (−0.866 − 0.499i)12-s + 5.81i·13-s + (−2.59 + 0.521i)14-s + 2.56i·15-s + (−0.5 + 0.866i)16-s + (−1.22 − 2.11i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.574 + 0.994i)5-s − 0.408i·6-s + (−0.660 − 0.750i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.405 + 0.703i)10-s + (−1.13 + 0.652i)11-s + (−0.249 − 0.144i)12-s + 1.61i·13-s + (−0.693 + 0.139i)14-s + 0.662i·15-s + (−0.125 + 0.216i)16-s + (−0.296 − 0.514i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0878 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0878 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.0878 - 0.996i$
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: \((2,\ 966,\ (\ :1/2),\ -0.0878 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.442149 + 0.482865i\)
\(L(\frac12)\) \(\approx\) \(0.442149 + 0.482865i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.74 + 1.98i)T \)
23 \( 1 + (-2.33 + 4.19i)T \)
good5 \( 1 + (1.28 - 2.22i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.74 - 2.16i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.81iT - 13T^{2} \)
17 \( 1 + (1.22 + 2.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.00 - 5.21i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 3.89T + 29T^{2} \)
31 \( 1 + (8.38 - 4.83i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.58 - 5.53i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 + 9.27iT - 43T^{2} \)
47 \( 1 + (-0.485 - 0.280i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.03 + 1.75i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.23 - 5.32i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.79 - 4.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.00 - 4.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + (-5.37 + 3.10i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.37 - 1.94i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.193T + 83T^{2} \)
89 \( 1 + (-0.225 + 0.389i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.63T + 97T^{2} \)
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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

−10.46504242712284083357487296183, −9.563209628853059670747628855077, −8.697462327597257132589652661852, −7.34666096487462441004067626732, −7.13736383946216051162626756801, −6.09176485888289895067448696035, −4.57405007987182871509481097713, −3.83009315632039404400397475717, −2.90670028156850638287390715928, −1.91717263512810713048017292759, 0.23627411399155204727498837110, 2.63087476657315358533593950690, 3.46348859582667304464944781408, 4.62357414777047938139062085488, 5.44027531547881234893218363950, 6.10320611764562601012029319945, 7.66909833413607908474364126052, 7.973123306428833124873903549790, 8.934919194340565156805412435510, 9.372572772296436948010195189372

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