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Properties

Label 2-966-7.2-c1-0-20
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 + (−1.94 + 3.37i)5-s + 0.999·6-s + (−1.32 − 2.29i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.94 + 3.37i)10-s + (−2.01 − 3.48i)11-s + (0.499 − 0.866i)12-s − 0.619·13-s − 2.64·14-s − 3.89·15-s + (−0.5 + 0.866i)16-s + (0.625 + 1.08i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.871 + 1.50i)5-s + 0.408·6-s + (−0.499 − 0.866i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.616 + 1.06i)10-s + (−0.606 − 1.05i)11-s + (0.144 − 0.249i)12-s − 0.171·13-s − 0.707·14-s − 1.00·15-s + (−0.125 + 0.216i)16-s + (0.151 + 0.262i)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: $\chi_{966} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.832 + 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166561 - 0.551135i\)
\(L(\frac12)\) \(\approx\) \(0.166561 - 0.551135i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.94 - 3.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.01 + 3.48i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.619T + 13T^{2} \)
17 \( 1 + (-0.625 - 1.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.20 + 7.28i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 9.41T + 29T^{2} \)
31 \( 1 + (2.59 + 4.49i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.690 - 1.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.41T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (-3.06 + 5.31i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.948 - 1.64i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.50 + 9.54i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.102 + 0.177i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.02 + 10.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.483T + 71T^{2} \)
73 \( 1 + (1.57 + 2.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.73 - 13.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.13T + 83T^{2} \)
89 \( 1 + (5.10 - 8.84i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−9.926793282772632334708475562889, −9.105116526130974837845224267819, −7.80775779796198846323538996226, −7.26531201351481620578429796511, −6.26124788561597663566000496435, −5.10583871142537140502801502132, −3.80958252426117253459501724341, −3.40250737749182928143221509044, −2.57397531417956955520802278409, −0.22152349875347294722002976660, 1.70125508240432751643241372498, 3.25205322202905297379961195260, 4.27970103985676189746838587581, 5.29074184604437333012909417488, 5.82260962659245006922337658075, 7.35553088378130526470136871781, 7.65502040662528951860202778214, 8.611692005321024158222630652253, 9.189032325590755533030067484370, 10.03753894996444202973803960140

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