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Properties

Label 2-966-161.10-c1-0-5
Degree $2$
Conductor $966$
Sign $-0.229 - 0.973i$
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 + (−1.32 + 0.126i)5-s + (0.989 − 0.142i)6-s + (−1.57 − 2.12i)7-s + (−0.841 + 0.540i)8-s + (−0.580 + 0.814i)9-s + (−0.313 + 1.29i)10-s + (−0.0601 + 0.0208i)11-s + (0.189 − 0.981i)12-s + (1.46 + 4.97i)13-s + (−2.52 + 0.795i)14-s + (−0.719 − 1.11i)15-s + (0.235 + 0.971i)16-s + (1.41 + 0.566i)17-s + ⋯
L(s)  = 1  + (0.231 − 0.668i)2-s + (0.264 + 0.513i)3-s + (−0.393 − 0.309i)4-s + (−0.592 + 0.0565i)5-s + (0.404 − 0.0580i)6-s + (−0.596 − 0.802i)7-s + (−0.297 + 0.191i)8-s + (−0.193 + 0.271i)9-s + (−0.0992 + 0.409i)10-s + (−0.0181 + 0.00627i)11-s + (0.0546 − 0.283i)12-s + (0.405 + 1.38i)13-s + (−0.674 + 0.212i)14-s + (−0.185 − 0.289i)15-s + (0.0589 + 0.242i)16-s + (0.343 + 0.137i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.229 - 0.973i$
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: \((2,\ 966,\ (\ :1/2),\ -0.229 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.382697 + 0.483607i\)
\(L(\frac12)\) \(\approx\) \(0.382697 + 0.483607i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 + 2.12i)T \)
23 \( 1 + (4.61 + 1.29i)T \)
good5 \( 1 + (1.32 - 0.126i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (0.0601 - 0.0208i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-1.46 - 4.97i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-1.41 - 0.566i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (7.07 - 2.83i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.882 - 6.14i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-2.86 + 0.136i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-1.11 - 0.795i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-5.75 - 2.63i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (0.982 - 1.52i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (5.22 - 3.01i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.71 + 3.90i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-5.49 - 1.33i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (3.74 + 1.93i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (0.0187 + 0.0971i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (7.84 + 9.05i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (10.0 - 12.7i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (9.76 - 10.2i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-2.26 - 4.95i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.686 + 14.4i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-1.03 + 2.27i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.26753805886594194576435103772, −9.660465658697125668422066623821, −8.705750932903079659555034923353, −7.950152701377492169085367347919, −6.77730499266671697323470880604, −5.97164555481855631579732441256, −4.35159493812603719809782448351, −4.14184761063467400339077958077, −3.13753721859331965801305557291, −1.72187214858528073015183783929, 0.24721962754087864081816723073, 2.39564498630762715357560719623, 3.42931375815569811496798989444, 4.46231863279575361116435136217, 5.82550810933320328074817226527, 6.17535648910555388754429248096, 7.33037321237662541102234082798, 8.133600914256778131621884978285, 8.568885288492113602285278637127, 9.600211299135008254939932679537

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