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Properties

Label 2-966-161.10-c1-0-27
Degree $2$
Conductor $966$
Sign $-0.900 + 0.435i$
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 + (1.68 − 0.161i)5-s + (−0.989 + 0.142i)6-s + (−0.672 − 2.55i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (0.399 − 1.64i)10-s + (3.81 − 1.31i)11-s + (−0.189 + 0.981i)12-s + (−0.831 − 2.83i)13-s + (−2.63 − 0.201i)14-s + (−0.916 − 1.42i)15-s + (0.235 + 0.971i)16-s + (2.93 + 1.17i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (−0.264 − 0.513i)3-s + (−0.393 − 0.309i)4-s + (0.754 − 0.0720i)5-s + (−0.404 + 0.0580i)6-s + (−0.254 − 0.967i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (0.126 − 0.521i)10-s + (1.14 − 0.397i)11-s + (−0.0546 + 0.283i)12-s + (−0.230 − 0.785i)13-s + (−0.705 − 0.0539i)14-s + (−0.236 − 0.368i)15-s + (0.0589 + 0.242i)16-s + (0.712 + 0.285i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.900 + 0.435i$
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: \((2,\ 966,\ (\ :1/2),\ -0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.366092 - 1.59641i\)
\(L(\frac12)\) \(\approx\) \(0.366092 - 1.59641i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.672 + 2.55i)T \)
23 \( 1 + (1.54 + 4.54i)T \)
good5 \( 1 + (-1.68 + 0.161i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-3.81 + 1.31i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (0.831 + 2.83i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-2.93 - 1.17i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (0.633 - 0.253i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.370 - 2.57i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-1.49 + 0.0713i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (4.13 + 2.94i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (1.76 + 0.805i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-2.52 + 3.92i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (0.394 - 0.227i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.22 + 5.47i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (10.6 + 2.57i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-11.1 - 5.73i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.820 - 4.25i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-7.18 - 8.29i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (5.83 - 7.41i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-5.95 + 6.24i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (4.67 + 10.2i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.0176 + 0.370i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (1.65 - 3.62i)T + (-63.5 - 73.3i)T^{2} \)
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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

−9.975843217940355175776243267785, −8.975514752636335754571452668374, −8.031371782639701225872118691000, −6.95197274608066849403859377240, −6.16259216846529563457484556962, −5.36356121880167865487740007978, −4.14742723328805101048355187053, −3.21483902365501257022292158762, −1.83895601451178212097388906614, −0.75453468003270610101362831473, 1.84997811399200594451732270481, 3.27198053089776028379344134041, 4.37789457198497824945870086153, 5.30795586691059175103876979911, 6.11427779745648018255379539497, 6.65032407547399233046143122426, 7.81979599820195764427126872073, 8.939619478367013937118808139014, 9.526734777723857199465231739370, 9.933760134188490575033356437534

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