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Properties

Label 2-966-69.68-c1-0-42
Degree $2$
Conductor $966$
Sign $-0.466 + 0.884i$
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 + (−1.69 + 0.349i)3-s − 4-s + 1.62·5-s + (−0.349 − 1.69i)6-s − i·7-s − i·8-s + (2.75 − 1.18i)9-s + 1.62i·10-s − 3.65·11-s + (1.69 − 0.349i)12-s + 0.597·13-s + 14-s + (−2.75 + 0.567i)15-s + 16-s − 3.33·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.979 + 0.201i)3-s − 0.5·4-s + 0.726·5-s + (−0.142 − 0.692i)6-s − 0.377i·7-s − 0.353i·8-s + (0.918 − 0.394i)9-s + 0.513i·10-s − 1.10·11-s + (0.489 − 0.100i)12-s + 0.165·13-s + 0.267·14-s + (−0.711 + 0.146i)15-s + 0.250·16-s − 0.808·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.466 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.466 + 0.884i$
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: \((2,\ 966,\ (\ :1/2),\ -0.466 + 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0466013 - 0.0772865i\)
\(L(\frac12)\) \(\approx\) \(0.0466013 - 0.0772865i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 0.349i)T \)
7 \( 1 + iT \)
23 \( 1 + (3.04 - 3.70i)T \)
good5 \( 1 - 1.62T + 5T^{2} \)
11 \( 1 + 3.65T + 11T^{2} \)
13 \( 1 - 0.597T + 13T^{2} \)
17 \( 1 + 3.33T + 17T^{2} \)
19 \( 1 - 0.263iT - 19T^{2} \)
29 \( 1 - 1.04iT - 29T^{2} \)
31 \( 1 + 8.47T + 31T^{2} \)
37 \( 1 + 7.11iT - 37T^{2} \)
41 \( 1 + 5.26iT - 41T^{2} \)
43 \( 1 - 0.223iT - 43T^{2} \)
47 \( 1 - 5.31iT - 47T^{2} \)
53 \( 1 + 4.55T + 53T^{2} \)
59 \( 1 - 6.54iT - 59T^{2} \)
61 \( 1 + 8.08iT - 61T^{2} \)
67 \( 1 + 12.2iT - 67T^{2} \)
71 \( 1 + 3.11iT - 71T^{2} \)
73 \( 1 + 7.94T + 73T^{2} \)
79 \( 1 + 8.69iT - 79T^{2} \)
83 \( 1 + 8.43T + 83T^{2} \)
89 \( 1 - 7.18T + 89T^{2} \)
97 \( 1 - 11.3iT - 97T^{2} \)
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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

−9.724280945541142008227741538319, −9.075281267069148499998728790597, −7.78276973906668531438838154263, −7.15873707675443634850394754847, −6.09804476557158220735466015364, −5.59644618706532647806648655398, −4.74430259440253617075988009586, −3.69232211434492051813143285123, −1.88520957647123520980108789216, −0.04513899209808702075920864376, 1.69948773260688498364187007795, 2.62707581685976998898172494834, 4.18204238104342092952237980607, 5.16355792264000681891204602307, 5.81085879572181052451366065054, 6.70052762297064219909305986776, 7.83801941034911461159212027660, 8.771660733225705146826639913532, 9.844589008068772955066042875432, 10.29708480192520788119132113693

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