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Properties

Label 2-966-21.5-c1-0-24
Degree $2$
Conductor $966$
Sign $-0.895 - 0.444i$
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  + (0.866 + 0.5i)2-s + (0.866 + 1.5i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 1.5i)5-s + 1.73i·6-s + (0.5 + 2.59i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (−1.5 + 0.866i)10-s + (−0.866 + 1.49i)12-s − 1.73i·13-s + (−0.866 + 2.5i)14-s − 3·15-s + (−0.5 + 0.866i)16-s + (−2.59 − 4.5i)17-s + (−2.59 + 1.5i)18-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.499 + 0.866i)3-s + (0.249 + 0.433i)4-s + (−0.387 + 0.670i)5-s + 0.707i·6-s + (0.188 + 0.981i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.474 + 0.273i)10-s + (−0.250 + 0.433i)12-s − 0.480i·13-s + (−0.231 + 0.668i)14-s − 0.774·15-s + (−0.125 + 0.216i)16-s + (−0.630 − 1.09i)17-s + (−0.612 + 0.353i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531159 + 2.26737i\)
\(L(\frac12)\) \(\approx\) \(0.531159 + 2.26737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (0.866 - 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + (2.59 + 4.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-2.59 + 4.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.79 - 4.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.19 - 9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12 - 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15iT - 71T^{2} \)
73 \( 1 + (-10.5 + 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.92T + 83T^{2} \)
89 \( 1 + (-3.46 + 6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−10.33525588395943019634214130189, −9.497978214839937994507537619770, −8.658358421293522651440055714986, −7.86449635899246022767096251320, −7.04532853718794092714058461838, −5.83777947186473117199178573516, −5.13468123834843549994647832446, −4.17198804798731653661667790111, −3.09772901026072418877457341082, −2.48057068822045702189526436727, 0.849761486407505170143592379115, 1.93806784612429135672285063979, 3.32627435883242887293893509564, 4.18216248992123602678516571441, 5.09215214477851443991482740471, 6.41268935276135310844838135633, 7.03154918378976038834858519640, 7.982660944608826613083464290765, 8.707878388822879816904561440834, 9.607912878219375262947230226615

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