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Properties

Label 2-966-7.2-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 + (−0.5 + 0.866i)5-s − 0.999·6-s + (−0.5 + 2.59i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 − 0.866i)10-s + (1.5 + 2.59i)11-s + (0.499 − 0.866i)12-s + 2·13-s + (−2 − 1.73i)14-s − 0.999·15-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s − 0.408·6-s + (−0.188 + 0.981i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (0.452 + 0.783i)11-s + (0.144 − 0.249i)12-s + 0.554·13-s + (−0.534 − 0.462i)14-s − 0.258·15-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)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: $\chi_{966} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0710195 + 1.11911i\)
\(L(\frac12)\) \(\approx\) \(0.0710195 + 1.11911i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 7T + 29T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.5 + 12.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5T + 97T^{2} \)
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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

−10.20320780789765225110446586581, −9.285192270259309433152250942349, −8.990464854579590107304891052584, −7.891537584264714559627735374903, −7.16716295638896830820786596010, −6.12137083800776495148739351435, −5.39287542387732173532036117338, −4.24916017516082147188017778464, −3.22012272079427468943625547674, −1.85660060654480228580367842629, 0.58384491166812376422974592835, 1.67031298452011834084374658265, 3.27550290087565972836322940897, 3.84109608454822358073086456610, 5.12200821235566040316104309564, 6.40974687769886499293627151330, 7.22034971420824515605337767721, 8.151611972253176742407187286930, 8.712364128495663211896807800994, 9.629451573649923335150980265556

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