Properties

Label 2-966-161.10-c1-0-31
Degree $2$
Conductor $966$
Sign $-0.869 + 0.494i$
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 + (3.08 − 0.294i)5-s + (0.989 − 0.142i)6-s + (−1.81 − 1.92i)7-s + (0.841 − 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.730 + 3.01i)10-s + (−4.47 + 1.55i)11-s + (−0.189 + 0.981i)12-s + (−1.80 − 6.14i)13-s + (2.41 − 1.08i)14-s + (−1.67 − 2.60i)15-s + (0.235 + 0.971i)16-s + (−2.17 − 0.868i)17-s + ⋯
L(s)  = 1  + (−0.231 + 0.668i)2-s + (−0.264 − 0.513i)3-s + (−0.393 − 0.309i)4-s + (1.37 − 0.131i)5-s + (0.404 − 0.0580i)6-s + (−0.686 − 0.727i)7-s + (0.297 − 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.230 + 0.952i)10-s + (−1.35 + 0.467i)11-s + (−0.0546 + 0.283i)12-s + (−0.500 − 1.70i)13-s + (0.644 − 0.290i)14-s + (−0.432 − 0.672i)15-s + (0.0589 + 0.242i)16-s + (−0.526 − 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.869 + 0.494i$
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: \((2,\ 966,\ (\ :1/2),\ -0.869 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0851934 - 0.322342i\)
\(L(\frac12)\) \(\approx\) \(0.0851934 - 0.322342i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 + 1.92i)T \)
23 \( 1 + (0.504 - 4.76i)T \)
good5 \( 1 + (-3.08 + 0.294i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (4.47 - 1.55i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (1.80 + 6.14i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (2.17 + 0.868i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (6.23 - 2.49i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.886 - 6.16i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (2.70 - 0.128i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-1.57 - 1.12i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (1.38 + 0.631i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-0.424 + 0.659i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-3.36 + 1.94i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.08 + 1.13i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (4.28 + 1.03i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (10.9 + 5.64i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.605 - 3.14i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-4.18 - 4.83i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (0.115 - 0.147i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-7.62 + 8.00i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (6.74 + 14.7i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.479 + 10.0i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-6.74 + 14.7i)T + (-63.5 - 73.3i)T^{2} \)
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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

−9.844297214838643402439442031074, −8.775645043332892767635849920630, −7.76809009489219820539299279528, −7.17805236736331685665026940349, −6.16241101605620764966173892063, −5.58779306599705207958463422436, −4.77729437260480066600890631491, −3.04981334126461069316187493141, −1.83628587006598749141541619742, −0.15426241725271164262152665578, 2.25389955281160289695445555754, 2.53052507745726638109357509718, 4.17445856871939507501321551107, 5.08409744205536640569115220901, 6.11188579065455817380980200974, 6.63592101115347052705926248232, 8.223872029209776485848986078308, 9.240357204738545528629387510671, 9.392762407223667323852671149301, 10.47942412318901704492750818370

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