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Properties

Label 2-966-7.2-c1-0-8
Degree $2$
Conductor $966$
Sign $-0.771 - 0.636i$
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  + (−0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−1.76 + 3.05i)5-s − 0.999·6-s + (1.80 − 1.92i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−1.76 − 3.05i)10-s + (2.57 + 4.46i)11-s + (0.499 − 0.866i)12-s + 6.48·13-s + (0.766 + 2.53i)14-s − 3.53·15-s + (−0.5 + 0.866i)16-s + (0.809 + 1.40i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.790 + 1.36i)5-s − 0.408·6-s + (0.684 − 0.729i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.558 − 0.967i)10-s + (0.776 + 1.34i)11-s + (0.144 − 0.249i)12-s + 1.79·13-s + (0.204 + 0.676i)14-s − 0.912·15-s + (−0.125 + 0.216i)16-s + (0.196 + 0.340i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.771 - 0.636i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.479602 + 1.33480i\)
\(L(\frac12)\) \(\approx\) \(0.479602 + 1.33480i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.80 + 1.92i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (1.76 - 3.05i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.57 - 4.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.48T + 13T^{2} \)
17 \( 1 + (-0.809 - 1.40i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 1.48T + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.53 - 9.58i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.06T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (4.65 - 8.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.71 + 9.90i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.06 + 12.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.48 + 7.76i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.71 - 6.43i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.2T + 71T^{2} \)
73 \( 1 + (-3.98 - 6.89i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.57 - 11.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.23T + 83T^{2} \)
89 \( 1 + (0.949 - 1.64i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.26173150855492501136897245338, −9.612133209928560533401403240011, −8.375892249598164011271763037144, −7.948337683786401625563565384607, −6.79837104751415573992654595754, −6.62703401921879787060489100358, −4.99783597549861892005804409388, −4.03106970516084724931715929257, −3.37440873583193259741872052205, −1.58847097716934519037079685854, 0.834863618472952651758885222794, 1.64546732750361612506961751520, 3.35187740026678515655165581272, 4.03650800042529299926421569515, 5.31981812525642297517196678096, 6.13204604566886025565350529080, 7.62164912952985061352545259810, 8.364661892834957036875972676285, 8.787044835049343961601150777081, 9.209372344629907129069127095607

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