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Properties

Label 2-966-69.68-c1-0-40
Degree $2$
Conductor $966$
Sign $0.374 + 0.927i$
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  + i·2-s + (0.656 − 1.60i)3-s − 4-s − 0.252·5-s + (1.60 + 0.656i)6-s − i·7-s − i·8-s + (−2.13 − 2.10i)9-s − 0.252i·10-s + 5.50·11-s + (−0.656 + 1.60i)12-s − 3.10·13-s + 14-s + (−0.165 + 0.404i)15-s + 16-s − 0.471·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.378 − 0.925i)3-s − 0.5·4-s − 0.112·5-s + (0.654 + 0.267i)6-s − 0.377i·7-s − 0.353i·8-s + (−0.712 − 0.701i)9-s − 0.0797i·10-s + 1.65·11-s + (−0.189 + 0.462i)12-s − 0.859·13-s + 0.267·14-s + (−0.0427 + 0.104i)15-s + 0.250·16-s − 0.114·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23493 - 0.833553i\)
\(L(\frac12)\) \(\approx\) \(1.23493 - 0.833553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.656 + 1.60i)T \)
7 \( 1 + iT \)
23 \( 1 + (-3.43 + 3.34i)T \)
good5 \( 1 + 0.252T + 5T^{2} \)
11 \( 1 - 5.50T + 11T^{2} \)
13 \( 1 + 3.10T + 13T^{2} \)
17 \( 1 + 0.471T + 17T^{2} \)
19 \( 1 + 2.38iT - 19T^{2} \)
29 \( 1 + 6.08iT - 29T^{2} \)
31 \( 1 - 1.40T + 31T^{2} \)
37 \( 1 + 5.87iT - 37T^{2} \)
41 \( 1 + 4.64iT - 41T^{2} \)
43 \( 1 + 5.09iT - 43T^{2} \)
47 \( 1 - 4.02iT - 47T^{2} \)
53 \( 1 - 1.95T + 53T^{2} \)
59 \( 1 - 3.53iT - 59T^{2} \)
61 \( 1 - 2.10iT - 61T^{2} \)
67 \( 1 - 5.88iT - 67T^{2} \)
71 \( 1 + 15.2iT - 71T^{2} \)
73 \( 1 + 3.34T + 73T^{2} \)
79 \( 1 + 13.9iT - 79T^{2} \)
83 \( 1 - 9.18T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 17.8iT - 97T^{2} \)
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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

−9.379886119455382872152704375369, −9.042940176403170445892413169050, −7.981524821943007936381951827558, −7.25640915969091316315431642597, −6.65021858416392991764821455754, −5.87032437426257213439646453793, −4.53666048605849823482079860879, −3.62133358947518560416009391144, −2.19068458720567157252759495155, −0.68822575600219736812261838811, 1.64300614849975218978937742710, 2.96548576705961399695303966230, 3.80045264057574668365952107785, 4.65527399405723844380632028067, 5.57470409912372129153732777008, 6.76754036225690939469887606381, 7.990667775814615633412118153417, 8.814225273095501857472850952547, 9.542661604659496132256916292153, 9.931295606933503918358724172096

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