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Properties

Label 2-966-161.10-c1-0-15
Degree $2$
Conductor $966$
Sign $0.865 + 0.500i$
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 + (−3.89 + 0.372i)5-s + (0.989 − 0.142i)6-s + (−2.62 + 0.292i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.923 + 3.80i)10-s + (5.29 − 1.83i)11-s + (0.189 − 0.981i)12-s + (−0.0556 − 0.189i)13-s + (−0.583 + 2.58i)14-s + (−2.11 − 3.29i)15-s + (0.235 + 0.971i)16-s + (−1.48 − 0.595i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (−1.74 + 0.166i)5-s + (0.404 − 0.0580i)6-s + (−0.993 + 0.110i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.291 + 1.20i)10-s + (1.59 − 0.552i)11-s + (0.0546 − 0.283i)12-s + (−0.0154 − 0.0525i)13-s + (−0.155 + 0.689i)14-s + (−0.546 − 0.850i)15-s + (0.0589 + 0.242i)16-s + (−0.360 − 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.865 + 0.500i$
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: \((2,\ 966,\ (\ :1/2),\ 0.865 + 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20856 - 0.324489i\)
\(L(\frac12)\) \(\approx\) \(1.20856 - 0.324489i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.62 - 0.292i)T \)
23 \( 1 + (-4.20 - 2.30i)T \)
good5 \( 1 + (3.89 - 0.372i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-5.29 + 1.83i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (0.0556 + 0.189i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (1.48 + 0.595i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-7.40 + 2.96i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.983 - 6.83i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-0.478 + 0.0227i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-6.70 - 4.77i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (9.27 + 4.23i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-5.21 + 8.12i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-1.70 + 0.986i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.70 + 4.93i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-5.06 - 1.22i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-2.86 - 1.47i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-1.68 - 8.73i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (3.03 + 3.50i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (6.72 - 8.55i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-7.19 + 7.54i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (2.88 + 6.32i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.217 - 4.56i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-1.99 + 4.37i)T + (-63.5 - 73.3i)T^{2} \)
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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

−9.962028856486895611435990045456, −9.040890246418729215936102607436, −8.694624907571121818057760088448, −7.33507613636236289587044691462, −6.72550323626639142515354453548, −5.31455173630166140177727968332, −4.25854643810388282338568275189, −3.43301164905920916766444374836, −3.10373807380300722268941382531, −0.827650402878404040468016457397, 0.879117721369973870750901687663, 3.09685633742279000729888924955, 3.86577560456468893108694461093, 4.58902429503769260336477427546, 6.10848501876624843006772105587, 6.85425239206380024714363291953, 7.47814298468810823956005972231, 8.162211251795384442256567818024, 9.140623791005516681265572732746, 9.702784346890600185963933655092

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