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Properties

Label 2-966-7.4-c1-0-18
Degree $2$
Conductor $966$
Sign $-0.0222 + 0.999i$
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 + (0.415 + 0.720i)5-s − 0.999·6-s + (2.64 − 0.108i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.415 − 0.720i)10-s + (1.22 − 2.12i)11-s + (0.499 + 0.866i)12-s − 5.30·13-s + (−1.41 − 2.23i)14-s + 0.831·15-s + (−0.5 − 0.866i)16-s + (1.64 − 2.84i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.185 + 0.322i)5-s − 0.408·6-s + (0.999 − 0.0410i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.131 − 0.227i)10-s + (0.370 − 0.641i)11-s + (0.144 + 0.249i)12-s − 1.47·13-s + (−0.378 − 0.597i)14-s + 0.214·15-s + (−0.125 − 0.216i)16-s + (0.398 − 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0222 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0222 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.0222 + 0.999i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.0222 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10416 - 1.12904i\)
\(L(\frac12)\) \(\approx\) \(1.10416 - 1.12904i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.108i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.415 - 0.720i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.22 + 2.12i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.30T + 13T^{2} \)
17 \( 1 + (-1.64 + 2.84i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.16 + 2.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.66T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (1.95 + 3.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.89 + 6.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.66 + 2.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.30 + 12.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.89 - 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.04T + 71T^{2} \)
73 \( 1 + (7.80 - 13.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.22 + 9.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.57T + 83T^{2} \)
89 \( 1 + (-6.47 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
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

−9.907143235710904020086277308978, −8.957711796131474295637529006456, −8.166671940180443663099246918699, −7.49115096420970036955879184005, −6.61585617970590668598954191137, −5.33722987058407675607509759272, −4.40609243767975411204762954338, −3.03680990732952671100643801529, −2.23510676972851782753612857139, −0.906834444287578961420523658176, 1.41027965192360130090467581800, 2.76416391596504629154821219503, 4.51108359416319296490181964056, 4.79756075720150382768695261852, 5.86141157022649437583838176004, 7.09929044487509971023395394647, 7.72326926258619256010362810201, 8.641835451519551065208640800630, 9.249154003356050663469587891821, 10.12141094514761106561099522651

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