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Properties

Label 2-966-7.2-c1-0-7
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 − 0.999·6-s + (−0.5 + 2.59i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)11-s + (−0.499 + 0.866i)12-s + 2·13-s + (2 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (1.5 + 2.59i)17-s + (0.499 + 0.866i)18-s + (−1 + 1.73i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.408·6-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.452 + 0.783i)11-s + (−0.144 + 0.249i)12-s + 0.554·13-s + (0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (0.363 + 0.630i)17-s + (0.117 + 0.204i)18-s + (−0.229 + 0.397i)19-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: $\chi_{966} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59044 - 0.100929i\)
\(L(\frac12)\) \(\approx\) \(1.59044 - 0.100929i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
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 \( 1 - 3T + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
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

−10.12030202280515934863219438291, −9.201008922910773703070888246982, −8.512463225007344424175332088643, −7.40853052139912933553449470450, −6.34345172898201077939508143062, −5.73672607528956596737959445817, −4.71631535687216859872883020456, −3.58796775581352829475431611223, −2.41397339127082877664149905780, −1.36775458321336203771993343710, 0.77498650709522305043262604468, 3.02371156951160400573092117696, 3.93691380783403132142678340115, 4.71817212245616034630759527867, 5.78426827727546476366644715165, 6.55444750928560972981546368585, 7.33353445921021949746536442260, 8.373323094762277666971478788471, 9.113226188481600547243663654726, 10.05854808830250924369635294119

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