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Properties

Label 2-966-161.68-c1-0-15
Degree $2$
Conductor $966$
Sign $0.520 - 0.854i$
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 + (−1.44 + 2.50i)5-s − 0.999i·6-s + (2.53 − 0.757i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−1.44 − 2.50i)10-s + (3.09 − 1.78i)11-s + (0.866 + 0.499i)12-s − 0.427i·13-s + (−0.611 + 2.57i)14-s − 2.89i·15-s + (−0.5 + 0.866i)16-s + (0.156 + 0.271i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.647 + 1.12i)5-s − 0.408i·6-s + (0.958 − 0.286i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.458 − 0.793i)10-s + (0.932 − 0.538i)11-s + (0.249 + 0.144i)12-s − 0.118i·13-s + (−0.163 + 0.687i)14-s − 0.748i·15-s + (−0.125 + 0.216i)16-s + (0.0380 + 0.0658i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.520 - 0.854i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.520 - 0.854i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.995838 + 0.559522i\)
\(L(\frac12)\) \(\approx\) \(0.995838 + 0.559522i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.53 + 0.757i)T \)
23 \( 1 + (3.63 + 3.12i)T \)
good5 \( 1 + (1.44 - 2.50i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.09 + 1.78i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.427iT - 13T^{2} \)
17 \( 1 + (-0.156 - 0.271i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.24 + 5.62i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 9.17T + 29T^{2} \)
31 \( 1 + (-7.49 + 4.32i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.85 + 3.95i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.7iT - 41T^{2} \)
43 \( 1 + 0.154iT - 43T^{2} \)
47 \( 1 + (-5.27 - 3.04i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.239 + 0.138i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.4 - 6.60i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.11 - 10.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.18 + 1.26i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + (-2.01 + 1.16i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.40 - 1.39i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.97T + 83T^{2} \)
89 \( 1 + (-2.07 + 3.58i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.5T + 97T^{2} \)
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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

−10.32861314330457622046209909241, −9.270098309955540940798374857413, −8.326899639016138403718740002444, −7.59286933721415124334990903136, −6.73098962210103841027563759950, −6.15533055641040614350034185488, −4.86742110315097805046416997219, −4.13472316714081506327591600856, −2.86067476278793020147035822174, −0.924019243643349420542558823692, 1.02070859847559982830609228609, 1.84886956027395197389050956560, 3.65565184341180917892223382874, 4.60336690473840877098557643494, 5.24131718036907085670881698784, 6.52721174068811504930724559887, 7.69911221190875729943794053332, 8.257965797284784771861601880739, 8.978319753738578204996696096194, 9.932598907854089807778168582478

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