Properties

Label 2-966-23.8-c1-0-10
Degree $2$
Conductor $966$
Sign $-0.902 - 0.431i$
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  + (−0.142 + 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (2.39 + 2.76i)5-s + (−0.959 + 0.281i)6-s + (0.841 + 0.540i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (−3.08 + 1.97i)10-s + (−0.108 − 0.754i)11-s + (−0.142 − 0.989i)12-s + (−0.436 + 0.280i)13-s + (−0.654 + 0.755i)14-s + (−1.52 + 3.33i)15-s + (0.841 + 0.540i)16-s + (−6.86 + 2.01i)17-s + ⋯
L(s)  = 1  + (−0.100 + 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (1.07 + 1.23i)5-s + (−0.391 + 0.115i)6-s + (0.317 + 0.204i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (−0.974 + 0.626i)10-s + (−0.0327 − 0.227i)11-s + (−0.0410 − 0.285i)12-s + (−0.121 + 0.0778i)13-s + (−0.175 + 0.201i)14-s + (−0.392 + 0.860i)15-s + (0.210 + 0.135i)16-s + (−1.66 + 0.488i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.902 - 0.431i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.902 - 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.393373 + 1.73522i\)
\(L(\frac12)\) \(\approx\) \(0.393373 + 1.73522i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.142 - 0.989i)T \)
3 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
23 \( 1 + (1.59 - 4.52i)T \)
good5 \( 1 + (-2.39 - 2.76i)T + (-0.711 + 4.94i)T^{2} \)
11 \( 1 + (0.108 + 0.754i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (0.436 - 0.280i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (6.86 - 2.01i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (-5.98 - 1.75i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-3.14 + 0.922i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (-1.74 + 3.82i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (-3.97 + 4.58i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (1.12 + 1.30i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-0.540 - 1.18i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + 5.35T + 47T^{2} \)
53 \( 1 + (-0.863 - 0.554i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-2.75 + 1.77i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-4.09 + 8.97i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (0.796 - 5.54i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (2.13 - 14.8i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (14.9 + 4.39i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (6.84 - 4.40i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-9.70 + 11.2i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (0.0160 + 0.0350i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (-2.90 - 3.34i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.10185817505664439599246675162, −9.619079866393529878667817283519, −8.786712390957419427210722890447, −7.79849864829171523305093387313, −6.92287684636751938505631237614, −6.07999702557616441325071977403, −5.43951909512172675495954580823, −4.24748660300045569542702359139, −3.06070007771898485454314132413, −1.97684136265553614292390833884, 0.851011627077253282748749602608, 1.88255949961571761461276669325, 2.84100862007465024149792696768, 4.52687649620613067397666983747, 5.00171077645363847130422216386, 6.17606144460087340725799387043, 7.18766480597824364950651107776, 8.358420180384947594591430560337, 8.879630488635239202875673337582, 9.606002421001701485064063401249

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