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Properties

Label 2-966-69.68-c1-0-18
Degree $2$
Conductor $966$
Sign $0.961 - 0.273i$
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 + (−0.529 + 1.64i)3-s − 4-s − 2.93·5-s + (−1.64 − 0.529i)6-s − i·7-s − i·8-s + (−2.43 − 1.74i)9-s − 2.93i·10-s − 2.76·11-s + (0.529 − 1.64i)12-s + 4.19·13-s + 14-s + (1.55 − 4.84i)15-s + 16-s − 1.10·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.305 + 0.952i)3-s − 0.5·4-s − 1.31·5-s + (−0.673 − 0.216i)6-s − 0.377i·7-s − 0.353i·8-s + (−0.813 − 0.581i)9-s − 0.929i·10-s − 0.832·11-s + (0.152 − 0.476i)12-s + 1.16·13-s + 0.267·14-s + (0.401 − 1.25i)15-s + 0.250·16-s − 0.268·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.961 - 0.273i$
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: \((2,\ 966,\ (\ :1/2),\ 0.961 - 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689299 + 0.0961685i\)
\(L(\frac12)\) \(\approx\) \(0.689299 + 0.0961685i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.529 - 1.64i)T \)
7 \( 1 + iT \)
23 \( 1 + (-2.65 - 3.99i)T \)
good5 \( 1 + 2.93T + 5T^{2} \)
11 \( 1 + 2.76T + 11T^{2} \)
13 \( 1 - 4.19T + 13T^{2} \)
17 \( 1 + 1.10T + 17T^{2} \)
19 \( 1 + 4.25iT - 19T^{2} \)
29 \( 1 - 0.175iT - 29T^{2} \)
31 \( 1 - 3.49T + 31T^{2} \)
37 \( 1 - 1.18iT - 37T^{2} \)
41 \( 1 + 6.45iT - 41T^{2} \)
43 \( 1 + 5.95iT - 43T^{2} \)
47 \( 1 - 6.79iT - 47T^{2} \)
53 \( 1 - 7.44T + 53T^{2} \)
59 \( 1 + 10.5iT - 59T^{2} \)
61 \( 1 + 5.65iT - 61T^{2} \)
67 \( 1 - 7.63iT - 67T^{2} \)
71 \( 1 + 5.46iT - 71T^{2} \)
73 \( 1 + 5.68T + 73T^{2} \)
79 \( 1 + 0.0580iT - 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 + 7.60T + 89T^{2} \)
97 \( 1 + 1.20iT - 97T^{2} \)
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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

−10.07273238232199535608315288731, −9.012459643025053533354176603690, −8.408571939827519704001029127998, −7.56663249048084997863206736969, −6.71191866090922077439053931368, −5.61723063742700297657208097646, −4.74512414784714982502101155239, −3.96501658137621310250750196510, −3.16903291693120247092603850289, −0.44081435350771085294573959889, 1.00139241827342335897270952908, 2.46060637561400786227888364138, 3.48342939207675281870008983351, 4.56704716883807985511292475260, 5.66064746326386139795579883515, 6.60211973276177275662008133850, 7.73232083902517820844692315436, 8.228662920782167957817909644735, 8.895247324968706085461151881021, 10.36476833594366682312868424645

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