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Properties

Label 2-966-161.45-c1-0-30
Degree $2$
Conductor $966$
Sign $-0.999 + 0.00264i$
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 + (−0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−2.02 − 3.51i)5-s − 0.999i·6-s + (2.03 − 1.69i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (2.02 − 3.51i)10-s + (−4.89 − 2.82i)11-s + (0.866 − 0.499i)12-s + 1.68i·13-s + (2.48 + 0.914i)14-s + 4.05i·15-s + (−0.5 − 0.866i)16-s + (−0.577 + 1.00i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.907 − 1.57i)5-s − 0.408i·6-s + (0.768 − 0.639i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.641 − 1.11i)10-s + (−1.47 − 0.852i)11-s + (0.249 − 0.144i)12-s + 0.467i·13-s + (0.663 + 0.244i)14-s + 1.04i·15-s + (−0.125 − 0.216i)16-s + (−0.140 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.999 + 0.00264i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.999 + 0.00264i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000257170 - 0.194758i\)
\(L(\frac12)\) \(\approx\) \(0.000257170 - 0.194758i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-2.03 + 1.69i)T \)
23 \( 1 + (3.50 - 3.27i)T \)
good5 \( 1 + (2.02 + 3.51i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.89 + 2.82i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.68iT - 13T^{2} \)
17 \( 1 + (0.577 - 1.00i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.27 - 5.67i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 6.42T + 29T^{2} \)
31 \( 1 + (2.49 + 1.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.89 + 1.09i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.70iT - 41T^{2} \)
43 \( 1 + 1.32iT - 43T^{2} \)
47 \( 1 + (-3.82 + 2.20i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.85 + 1.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.54 - 3.77i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.20 + 9.01i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.92 + 2.84i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.18T + 71T^{2} \)
73 \( 1 + (9.62 + 5.55i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.79 - 5.65i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.847T + 83T^{2} \)
89 \( 1 + (-6.80 - 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.92T + 97T^{2} \)
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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

−9.402107218531305300194502415491, −8.321171253394295119236641646573, −7.87870679929413119215934822512, −7.40004502056852994474926063474, −5.76777965499664261719685021369, −5.35159313500757063973359727631, −4.43101869530829794889088368159, −3.65854008426030957425730138983, −1.51871746309436473276798846355, −0.085312753696458930406976824411, 2.33998760893840607445505012009, 2.99063817465912141026903256859, 4.23724721548718180153743366208, 5.08099066540407859211238029399, 5.94491736174849960763922674228, 7.25763210342472172031967382100, 7.62772074674232389878220629309, 8.874070728206083892597879857705, 10.10167896959606210378345373732, 10.55998914809802861981936371680

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