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Properties

Label 2-966-7.4-c1-0-24
Degree $2$
Conductor $966$
Sign $0.832 + 0.553i$
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 + (0.239 + 0.414i)5-s + 0.999·6-s + (1.32 − 2.29i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.239 + 0.414i)10-s + (2.67 − 4.63i)11-s + (0.499 + 0.866i)12-s − 4.70·13-s + 2.64·14-s + 0.479·15-s + (−0.5 − 0.866i)16-s + (1.08 − 1.87i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.107 + 0.185i)5-s + 0.408·6-s + (0.499 − 0.866i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.0757 + 0.131i)10-s + (0.806 − 1.39i)11-s + (0.144 + 0.249i)12-s − 1.30·13-s + 0.707·14-s + 0.123·15-s + (−0.125 − 0.216i)16-s + (0.262 − 0.455i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.832 + 0.553i$
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: \((2,\ 966,\ (\ :1/2),\ 0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97806 - 0.597800i\)
\(L(\frac12)\) \(\approx\) \(1.97806 - 0.597800i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 2.29i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.239 - 0.414i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.67 + 4.63i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.70T + 13T^{2} \)
17 \( 1 + (-1.08 + 1.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.87 - 3.24i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 4.74T + 29T^{2} \)
31 \( 1 + (-4.88 + 8.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.35 - 2.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.253T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-4.96 - 8.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.23 - 2.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.27 - 10.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.47 - 7.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.35 + 5.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.774T + 71T^{2} \)
73 \( 1 + (-5.61 + 9.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.423 + 0.733i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + (-1.60 - 2.78i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−9.794764687473023272099962010235, −8.970500049290910121703465271737, −7.87376301275339498504645850940, −7.57899683924729223010809041799, −6.52088228702524545624051327219, −5.82317231631330017106580739755, −4.64329734720957712675561044896, −3.71086393408800221329302336783, −2.59124759734991352928210211836, −0.873936767804901149667619774306, 1.70138446765096994848754670551, 2.59865715959141088111168215992, 3.82072166885479504917372188920, 4.95108684530485631835554858415, 5.20045251484526634147659668468, 6.68442673260403368557102748074, 7.61962476975478077310291664283, 8.779440764958057187992600215249, 9.435373264992443464850710967400, 9.929719802245123484411599254265

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