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Properties

Label 2-966-161.10-c1-0-0
Degree $2$
Conductor $966$
Sign $-0.421 + 0.906i$
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 + (−2.51 + 0.239i)5-s + (−0.989 + 0.142i)6-s + (2.17 − 1.51i)7-s + (0.841 − 0.540i)8-s + (−0.580 + 0.814i)9-s + (0.594 − 2.45i)10-s + (−0.754 + 0.261i)11-s + (0.189 − 0.981i)12-s + (0.889 + 3.03i)13-s + (0.716 + 2.54i)14-s + (−1.36 − 2.12i)15-s + (0.235 + 0.971i)16-s + (−6.85 − 2.74i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (−1.12 + 0.107i)5-s + (−0.404 + 0.0580i)6-s + (0.821 − 0.570i)7-s + (0.297 − 0.191i)8-s + (−0.193 + 0.271i)9-s + (0.188 − 0.775i)10-s + (−0.227 + 0.0787i)11-s + (0.0546 − 0.283i)12-s + (0.246 + 0.840i)13-s + (0.191 + 0.680i)14-s + (−0.352 − 0.547i)15-s + (0.0589 + 0.242i)16-s + (−1.66 − 0.665i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.421 + 0.906i$
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: \((2,\ 966,\ (\ :1/2),\ -0.421 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0130278 - 0.0204234i\)
\(L(\frac12)\) \(\approx\) \(0.0130278 - 0.0204234i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 1.51i)T \)
23 \( 1 + (3.78 - 2.94i)T \)
good5 \( 1 + (2.51 - 0.239i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (0.754 - 0.261i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.889 - 3.03i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (6.85 + 2.74i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (2.26 - 0.907i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.927 + 6.45i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (3.97 - 0.189i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-2.74 - 1.95i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (7.54 + 3.44i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-2.62 + 4.07i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-1.06 + 0.615i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.12 + 7.47i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-13.0 - 3.16i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (8.22 + 4.23i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.774 - 4.01i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-4.21 - 4.86i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (3.46 - 4.40i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (2.15 - 2.26i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (6.22 + 13.6i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.520 - 10.9i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (0.0690 - 0.151i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.56718785743210924229035444851, −9.606142035083185641427469524740, −8.706787160168806370177849215260, −8.092457588104762865211556813698, −7.35684704084682094831644805808, −6.58532574784710232981833677500, −5.24588153484382994554351094347, −4.25178563450256230239425039277, −3.91520009279023956682487232878, −2.07180879990883515356106983725, 0.01138500102798022666926177338, 1.70283406754368056476345361478, 2.77180636351944915123189090224, 3.94623045464538667595225375798, 4.76357508551753658395132769759, 6.01149236688369309659554728791, 7.18435062761821996024556884488, 8.198604750442093278386726479958, 8.366913772600472209186351412614, 9.237295359721129943287441491387

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