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Properties

Label 2-966-161.68-c1-0-28
Degree $2$
Conductor $966$
Sign $-0.898 + 0.439i$
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  + (0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (2.02 − 3.51i)5-s + 0.999i·6-s + (−2.03 − 1.69i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−2.02 − 3.51i)10-s + (4.89 − 2.82i)11-s + (0.866 + 0.499i)12-s − 1.68i·13-s + (−2.48 + 0.914i)14-s + 4.05i·15-s + (−0.5 + 0.866i)16-s + (0.577 + 1.00i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.907 − 1.57i)5-s + 0.408i·6-s + (−0.768 − 0.639i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.641 − 1.11i)10-s + (1.47 − 0.852i)11-s + (0.249 + 0.144i)12-s − 0.467i·13-s + (−0.663 + 0.244i)14-s + 1.04i·15-s + (−0.125 + 0.216i)16-s + (0.140 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.898 + 0.439i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.898 + 0.439i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.363653 - 1.57204i\)
\(L(\frac12)\) \(\approx\) \(0.363653 - 1.57204i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.03 + 1.69i)T \)
23 \( 1 + (-4.58 - 1.39i)T \)
good5 \( 1 + (-2.02 + 3.51i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.89 + 2.82i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.68iT - 13T^{2} \)
17 \( 1 + (-0.577 - 1.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.27 - 5.67i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 6.42T + 29T^{2} \)
31 \( 1 + (2.49 - 1.44i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.89 + 1.09i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.70iT - 41T^{2} \)
43 \( 1 + 1.32iT - 43T^{2} \)
47 \( 1 + (-3.82 - 2.20i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.85 + 1.07i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.54 + 3.77i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.20 + 9.01i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.92 + 2.84i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.18T + 71T^{2} \)
73 \( 1 + (9.62 - 5.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.79 - 5.65i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.847T + 83T^{2} \)
89 \( 1 + (6.80 - 11.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.92T + 97T^{2} \)
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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

−9.634045279098228948810197067609, −9.161988179453961097893358763212, −8.359465501849796855341958938080, −6.77716835440219426086099056248, −5.84026074663042527087765001524, −5.41100084326935079441080741818, −4.14262281108298508634444605568, −3.59965727951052661698061345662, −1.68070386035621751472177509296, −0.73327123319072722761703328541, 2.03863530942752147965952519263, 3.02406597502028535529484959764, 4.24424224458096771376517057258, 5.53160096626285739290845903321, 6.29193912853577902050711582663, 6.93913443343717693269550108075, 7.12819745796521046750774066630, 8.967344996439862010014214871480, 9.424160021312621524727201391836, 10.32181703882427093903440672097

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