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Properties

Label 2-966-161.45-c1-0-31
Degree $2$
Conductor $966$
Sign $-0.331 - 0.943i$
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 + (0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (−2.07 − 3.58i)5-s − 0.999i·6-s + (−1.78 − 1.95i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−2.07 + 3.58i)10-s + (−2.94 − 1.70i)11-s + (−0.866 + 0.499i)12-s + 0.0118i·13-s + (−0.795 + 2.52i)14-s − 4.14i·15-s + (−0.5 − 0.866i)16-s + (−3.74 + 6.48i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.926 − 1.60i)5-s − 0.408i·6-s + (−0.675 − 0.737i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.654 + 1.13i)10-s + (−0.889 − 0.513i)11-s + (−0.249 + 0.144i)12-s + 0.00329i·13-s + (−0.212 + 0.674i)14-s − 1.06i·15-s + (−0.125 − 0.216i)16-s + (−0.908 + 1.57i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.331 - 0.943i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0569021 + 0.0803140i\)
\(L(\frac12)\) \(\approx\) \(0.0569021 + 0.0803140i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (1.78 + 1.95i)T \)
23 \( 1 + (-2.42 + 4.13i)T \)
good5 \( 1 + (2.07 + 3.58i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.94 + 1.70i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.0118iT - 13T^{2} \)
17 \( 1 + (3.74 - 6.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.10 - 3.64i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 3.05T + 29T^{2} \)
31 \( 1 + (-5.66 - 3.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.217 + 0.125i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.42iT - 41T^{2} \)
43 \( 1 - 6.14iT - 43T^{2} \)
47 \( 1 + (-1.66 + 0.963i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.02 - 3.48i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.24 + 2.44i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.73 + 3.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.67 + 3.27i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 + (12.4 + 7.20i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.9 - 8.03i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.21T + 83T^{2} \)
89 \( 1 + (2.47 + 4.28i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.85T + 97T^{2} \)
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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

−9.324746666308612310765959012226, −8.570535377714964445317375115777, −8.167873803839385372055106370542, −7.29620064515568906146326487789, −5.81406115486669298293735645036, −4.55942286012256844305955256707, −4.03440331266065266731538168661, −3.06932468132915363390207370096, −1.40361425938169745333627431405, −0.04921150836300893251542522688, 2.59286413075512658393503945782, 2.98582308109095732047021287006, 4.42048230844549071436912985445, 5.67006142065417387012220091975, 6.82355428379315857914906844826, 7.16809421792618297654518041453, 7.82381085307618978898896359947, 8.900449253751757496108388165913, 9.643145764022914185298326980673, 10.42320085475006188429894697151

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