Properties

Label 2-966-161.2-c1-0-14
Degree $2$
Conductor $966$
Sign $0.999 + 0.00443i$
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  + (0.888 − 0.458i)2-s + (−0.786 + 0.618i)3-s + (0.580 − 0.814i)4-s + (0.468 − 0.446i)5-s + (−0.415 + 0.909i)6-s + (2.25 + 1.39i)7-s + (0.142 − 0.989i)8-s + (0.235 − 0.971i)9-s + (0.211 − 0.612i)10-s + (5.34 + 2.75i)11-s + (0.0475 + 0.998i)12-s + (−1.24 − 1.43i)13-s + (2.63 + 0.205i)14-s + (−0.0921 + 0.641i)15-s + (−0.327 − 0.945i)16-s + (−3.72 + 0.355i)17-s + ⋯
L(s)  = 1  + (0.628 − 0.324i)2-s + (−0.453 + 0.356i)3-s + (0.290 − 0.407i)4-s + (0.209 − 0.199i)5-s + (−0.169 + 0.371i)6-s + (0.850 + 0.525i)7-s + (0.0503 − 0.349i)8-s + (0.0785 − 0.323i)9-s + (0.0669 − 0.193i)10-s + (1.61 + 0.830i)11-s + (0.0137 + 0.288i)12-s + (−0.344 − 0.397i)13-s + (0.704 + 0.0548i)14-s + (−0.0237 + 0.165i)15-s + (−0.0817 − 0.236i)16-s + (−0.902 + 0.0861i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.999 + 0.00443i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.999 + 0.00443i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37266 - 0.00526643i\)
\(L(\frac12)\) \(\approx\) \(2.37266 - 0.00526643i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.888 + 0.458i)T \)
3 \( 1 + (0.786 - 0.618i)T \)
7 \( 1 + (-2.25 - 1.39i)T \)
23 \( 1 + (1.05 - 4.67i)T \)
good5 \( 1 + (-0.468 + 0.446i)T + (0.237 - 4.99i)T^{2} \)
11 \( 1 + (-5.34 - 2.75i)T + (6.38 + 8.96i)T^{2} \)
13 \( 1 + (1.24 + 1.43i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (3.72 - 0.355i)T + (16.6 - 3.21i)T^{2} \)
19 \( 1 + (1.41 + 0.135i)T + (18.6 + 3.59i)T^{2} \)
29 \( 1 + (-3.91 + 8.57i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-7.18 + 2.87i)T + (22.4 - 21.3i)T^{2} \)
37 \( 1 + (-0.899 + 3.70i)T + (-32.8 - 16.9i)T^{2} \)
41 \( 1 + (-4.99 - 1.46i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-1.36 - 9.48i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (-1.62 + 2.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-9.04 - 1.74i)T + (49.2 + 19.6i)T^{2} \)
59 \( 1 + (-4.48 + 12.9i)T + (-46.3 - 36.4i)T^{2} \)
61 \( 1 + (1.83 + 1.44i)T + (14.3 + 59.2i)T^{2} \)
67 \( 1 + (0.269 - 5.65i)T + (-66.6 - 6.36i)T^{2} \)
71 \( 1 + (8.71 + 5.60i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (5.41 - 7.60i)T + (-23.8 - 68.9i)T^{2} \)
79 \( 1 + (5.49 - 1.05i)T + (73.3 - 29.3i)T^{2} \)
83 \( 1 + (5.08 - 1.49i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (13.4 + 5.36i)T + (64.4 + 61.4i)T^{2} \)
97 \( 1 + (8.55 + 2.51i)T + (81.6 + 52.4i)T^{2} \)
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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

−9.924805041066475167711064780267, −9.478089460984287706637788823883, −8.480800062581502406455393359669, −7.32825709803132759793480029840, −6.31602282939782010585451830121, −5.61388382017580381163708686737, −4.52174599075153944081248787780, −4.13184565123653458835644025559, −2.49943710400375220277986949787, −1.37812493138922407994730425821, 1.19501186266782820436253139081, 2.56644894047483498696191924051, 4.09471463389034621514747594149, 4.61938206010843848632735705140, 5.83280391310446313289460293222, 6.68523096463137416808184368955, 7.04056333437659495727559907109, 8.407707058415558561924097205166, 8.846899731461992731988506945535, 10.35829123923814985586642272005

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