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Properties

Label 2-966-161.10-c1-0-11
Degree $2$
Conductor $966$
Sign $0.146 - 0.989i$
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 + (−1.63 + 0.156i)5-s + (−0.989 + 0.142i)6-s + (−0.689 − 2.55i)7-s + (0.841 − 0.540i)8-s + (−0.580 + 0.814i)9-s + (0.387 − 1.59i)10-s + (2.48 − 0.859i)11-s + (0.189 − 0.981i)12-s + (0.647 + 2.20i)13-s + (2.63 + 0.183i)14-s + (−0.888 − 1.38i)15-s + (0.235 + 0.971i)16-s + (1.98 + 0.795i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (−0.731 + 0.0698i)5-s + (−0.404 + 0.0580i)6-s + (−0.260 − 0.965i)7-s + (0.297 − 0.191i)8-s + (−0.193 + 0.271i)9-s + (0.122 − 0.505i)10-s + (0.748 − 0.259i)11-s + (0.0546 − 0.283i)12-s + (0.179 + 0.611i)13-s + (0.705 + 0.0490i)14-s + (−0.229 − 0.356i)15-s + (0.0589 + 0.242i)16-s + (0.482 + 0.192i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.146 - 0.989i$
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: \((2,\ 966,\ (\ :1/2),\ 0.146 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.970431 + 0.837529i\)
\(L(\frac12)\) \(\approx\) \(0.970431 + 0.837529i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.689 + 2.55i)T \)
23 \( 1 + (-4.61 - 1.31i)T \)
good5 \( 1 + (1.63 - 0.156i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-2.48 + 0.859i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.647 - 2.20i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-1.98 - 0.795i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-1.53 + 0.616i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-1.10 - 7.65i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-5.68 + 0.270i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (1.22 + 0.873i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-7.20 - 3.28i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (3.37 - 5.25i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-9.53 + 5.50i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.68 - 4.91i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (6.85 + 1.66i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (2.19 + 1.13i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.173 - 0.902i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-8.05 - 9.29i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-2.94 + 3.73i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (1.28 - 1.34i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (2.18 + 4.79i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.521 + 10.9i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-3.10 + 6.78i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.05621254880222483584783619797, −9.263382550981303211640928232454, −8.567608112278383640436468739406, −7.59337901327504991928974839091, −7.01885407631696375842887670733, −6.06981345103668434002590714368, −4.81835540996464076663644545133, −4.01325136248227037329616399989, −3.24531757031976636951450995319, −1.09573763596627927241652883158, 0.822150829204916605794371922646, 2.31693116344996354129632290322, 3.24059060800164518629920918911, 4.23342431244343478878584270562, 5.48406840643628321847141555276, 6.47658272503214532726010216498, 7.58467775555632024249800439179, 8.217479086533986405307424240701, 9.044652082715694500450223845402, 9.690056893466037340663293392557

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