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Properties

Label 2-966-161.10-c1-0-12
Degree $2$
Conductor $966$
Sign $-0.579 - 0.814i$
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  + (−0.327 + 0.945i)2-s + (0.458 + 0.888i)3-s + (−0.786 − 0.618i)4-s + (0.301 − 0.0288i)5-s + (−0.989 + 0.142i)6-s + (−0.249 + 2.63i)7-s + (0.841 − 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.0714 + 0.294i)10-s + (3.73 − 1.29i)11-s + (0.189 − 0.981i)12-s + (0.539 + 1.83i)13-s + (−2.40 − 1.09i)14-s + (0.163 + 0.254i)15-s + (0.235 + 0.971i)16-s + (3.12 + 1.25i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (0.134 − 0.0128i)5-s + (−0.404 + 0.0580i)6-s + (−0.0944 + 0.995i)7-s + (0.297 − 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.0225 + 0.0931i)10-s + (1.12 − 0.389i)11-s + (0.0546 − 0.283i)12-s + (0.149 + 0.509i)13-s + (−0.643 − 0.293i)14-s + (0.0423 + 0.0658i)15-s + (0.0589 + 0.242i)16-s + (0.757 + 0.303i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.579 - 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.579 - 0.814i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.579 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.709996 + 1.37634i\)
\(L(\frac12)\) \(\approx\) \(0.709996 + 1.37634i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.327 - 0.945i)T \)
3 \( 1 + (-0.458 - 0.888i)T \)
7 \( 1 + (0.249 - 2.63i)T \)
23 \( 1 + (3.17 - 3.59i)T \)
good5 \( 1 + (-0.301 + 0.0288i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-3.73 + 1.29i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.539 - 1.83i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-3.12 - 1.25i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-7.95 + 3.18i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.182 - 1.26i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (9.93 - 0.473i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-3.33 - 2.37i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-2.25 - 1.03i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-2.60 + 4.05i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (9.73 - 5.61i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.27 - 7.62i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-7.18 - 1.74i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (5.36 + 2.76i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.294 - 1.53i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-4.04 - 4.67i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-0.581 + 0.739i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (6.89 - 7.23i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-4.53 - 9.93i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.299 - 6.29i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-7.30 + 15.9i)T + (-63.5 - 73.3i)T^{2} \)
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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

−9.750586023507702009595575827329, −9.470936065035187957018659342451, −8.796083870956303661085256889625, −7.86149868675810298551745556798, −6.97442477061786258722338477487, −5.80563788706914994466969167726, −5.44417063103865652265726163675, −4.08774921152955187812102881238, −3.16575051864059297332628764599, −1.56029148559038653978506161673, 0.839727631655079285058932044915, 1.89213651213315427646016718696, 3.38837229891905904620775558061, 3.94931956271532885461736624197, 5.34748829190727193432093552678, 6.42652508349402057994619223084, 7.52768030800214936964048111393, 7.85216457287500443831935697977, 9.131827706063689074479124536615, 9.788116220494236881735461934812

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