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Properties

Label 2-966-161.10-c1-0-14
Degree $2$
Conductor $966$
Sign $0.629 + 0.776i$
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 + (−1.97 + 0.188i)5-s + (−0.989 + 0.142i)6-s + (−0.675 + 2.55i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.467 + 1.92i)10-s + (2.38 − 0.826i)11-s + (−0.189 + 0.981i)12-s + (0.570 + 1.94i)13-s + (2.19 + 1.47i)14-s + (1.07 + 1.66i)15-s + (0.235 + 0.971i)16-s + (3.52 + 1.41i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (−0.264 − 0.513i)3-s + (−0.393 − 0.309i)4-s + (−0.882 + 0.0843i)5-s + (−0.404 + 0.0580i)6-s + (−0.255 + 0.966i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.147 + 0.609i)10-s + (0.719 − 0.249i)11-s + (−0.0546 + 0.283i)12-s + (0.158 + 0.538i)13-s + (0.587 + 0.394i)14-s + (0.276 + 0.430i)15-s + (0.0589 + 0.242i)16-s + (0.856 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.629 + 0.776i$
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: \((2,\ 966,\ (\ :1/2),\ 0.629 + 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19029 - 0.567362i\)
\(L(\frac12)\) \(\approx\) \(1.19029 - 0.567362i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.675 - 2.55i)T \)
23 \( 1 + (-2.28 + 4.21i)T \)
good5 \( 1 + (1.97 - 0.188i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-2.38 + 0.826i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.570 - 1.94i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-3.52 - 1.41i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-4.92 + 1.97i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.905 + 6.29i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-0.745 + 0.0355i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (0.924 + 0.658i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-10.0 - 4.58i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (2.01 - 3.12i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (1.95 - 1.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.02 - 6.31i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-8.15 - 1.97i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-7.45 - 3.84i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-1.65 - 8.59i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (5.71 + 6.59i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-3.00 + 3.82i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-6.24 + 6.55i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-7.56 - 16.5i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.832 + 17.4i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (0.578 - 1.26i)T + (-63.5 - 73.3i)T^{2} \)
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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

−9.889334539354473885170135822536, −9.131886692404763804322357639984, −8.300587410512099475955310083312, −7.41293818721344955253780011945, −6.34262043134195637168091905958, −5.60429413527479801995137751401, −4.44209608957512951630157923140, −3.45306353966649917004471131895, −2.43308518239592368868532918198, −0.954186430479940530279806637702, 0.869865639726113447999708623039, 3.49739171646864283746447508219, 3.73995773924846783384398651816, 4.93934814535067744137021991821, 5.69928475845703962917757290248, 6.94629749769757071770165024993, 7.45566978361457130425923938505, 8.272754098308042510963291920754, 9.388423394930079841426306643720, 9.983956036869691893853387922376

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