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Properties

Label 2-966-69.68-c1-0-23
Degree $2$
Conductor $966$
Sign $0.997 - 0.0721i$
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 + (0.696 + 1.58i)3-s − 4-s + 3.16·5-s + (1.58 − 0.696i)6-s − i·7-s + i·8-s + (−2.02 + 2.20i)9-s − 3.16i·10-s + 1.43·11-s + (−0.696 − 1.58i)12-s + 2.73·13-s − 14-s + (2.20 + 5.01i)15-s + 16-s + 0.955·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.402 + 0.915i)3-s − 0.5·4-s + 1.41·5-s + (0.647 − 0.284i)6-s − 0.377i·7-s + 0.353i·8-s + (−0.676 + 0.736i)9-s − 0.999i·10-s + 0.433·11-s + (−0.201 − 0.457i)12-s + 0.759·13-s − 0.267·14-s + (0.568 + 1.29i)15-s + 0.250·16-s + 0.231·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.997 - 0.0721i$
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: \((2,\ 966,\ (\ :1/2),\ 0.997 - 0.0721i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26760 + 0.0819035i\)
\(L(\frac12)\) \(\approx\) \(2.26760 + 0.0819035i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.696 - 1.58i)T \)
7 \( 1 + iT \)
23 \( 1 + (-1.60 + 4.51i)T \)
good5 \( 1 - 3.16T + 5T^{2} \)
11 \( 1 - 1.43T + 11T^{2} \)
13 \( 1 - 2.73T + 13T^{2} \)
17 \( 1 - 0.955T + 17T^{2} \)
19 \( 1 - 7.49iT - 19T^{2} \)
29 \( 1 + 8.82iT - 29T^{2} \)
31 \( 1 + 2.94T + 31T^{2} \)
37 \( 1 - 2.44iT - 37T^{2} \)
41 \( 1 - 3.05iT - 41T^{2} \)
43 \( 1 - 2.41iT - 43T^{2} \)
47 \( 1 - 8.46iT - 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 - 4.28iT - 59T^{2} \)
61 \( 1 - 3.13iT - 61T^{2} \)
67 \( 1 + 2.67iT - 67T^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 + 4.14T + 73T^{2} \)
79 \( 1 + 4.98iT - 79T^{2} \)
83 \( 1 - 5.69T + 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 - 4.71iT - 97T^{2} \)
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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

−10.07467951615792981947530761850, −9.467009397017182183382080779266, −8.693661206045980218011197415268, −7.84531307985317078798219739057, −6.22824454803025456142587657708, −5.68343153742820614670767579430, −4.49637323496801646060945185683, −3.68754618887824704841033364441, −2.57469528900681107384788302416, −1.48156614431479997971202909819, 1.22370601444659766669114910391, 2.35808861248064272924098716261, 3.55902958845159766527455625129, 5.27123558497063215929330824949, 5.74780261349114782226765739936, 6.79468591280047073489055401995, 7.12874783496916025972149576064, 8.532784762702217129247931077649, 8.989582357310468819611459217305, 9.586931640485234820994518321324

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