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Properties

Label 2-966-7.4-c1-0-3
Degree $2$
Conductor $966$
Sign $-0.991 + 0.126i$
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 + (−0.780 − 1.35i)5-s − 0.999·6-s + (0.5 + 2.59i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.780 − 1.35i)10-s + (−1.5 + 2.59i)11-s + (−0.499 − 0.866i)12-s + 5.12·13-s + (−2 + 1.73i)14-s + 1.56·15-s + (−0.5 − 0.866i)16-s + (−3.84 + 6.65i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.349 − 0.604i)5-s − 0.408·6-s + (0.188 + 0.981i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.246 − 0.427i)10-s + (−0.452 + 0.783i)11-s + (−0.144 − 0.249i)12-s + 1.42·13-s + (−0.534 + 0.462i)14-s + 0.403·15-s + (−0.125 − 0.216i)16-s + (−0.931 + 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.991 + 0.126i$
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: \((2,\ 966,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0614574 - 0.968440i\)
\(L(\frac12)\) \(\approx\) \(0.0614574 - 0.968440i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.780 + 1.35i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.12T + 13T^{2} \)
17 \( 1 + (3.84 - 6.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.56 + 4.43i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 8.12T + 29T^{2} \)
31 \( 1 + (5.34 - 9.25i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.56 + 7.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.12T + 41T^{2} \)
43 \( 1 + 4.24T + 43T^{2} \)
47 \( 1 + (-2.28 - 3.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.78 - 3.08i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.78 - 4.81i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.56T + 71T^{2} \)
73 \( 1 + (-5.96 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.18 - 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.43T + 83T^{2} \)
89 \( 1 + (-4.12 - 7.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.684T + 97T^{2} \)
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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

−10.75618896253632905201904958168, −9.186902671834959843995297034311, −8.810690691234572569023430409880, −8.125871147777217239232688776265, −6.90047060177718909233198804692, −6.02265462178953227187472094271, −5.25265086422370415470923845168, −4.42757315066369881782894070906, −3.57725880496013843522034315143, −1.96099803363809023124931215359, 0.40519644280904451749285082641, 1.85384854996235994534110290183, 3.26008737015181042572538406773, 3.95168450934028951524638683919, 5.17343921830887270756629001507, 6.15710070658001845151138145240, 6.98337200124481083298114474751, 7.82225966398850858871214608755, 8.723614727967564457759334978473, 9.824139020631897647069576692706

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