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Properties

Label 2-966-161.68-c1-0-17
Degree $2$
Conductor $966$
Sign $0.999 - 0.0155i$
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 + (−0.242 + 0.420i)5-s + 0.999i·6-s + (−2.51 − 0.834i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−0.242 − 0.420i)10-s + (2.65 − 1.53i)11-s + (−0.866 − 0.499i)12-s + 5.07i·13-s + (1.97 − 1.75i)14-s + 0.485i·15-s + (−0.5 + 0.866i)16-s + (0.418 + 0.725i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.108 + 0.188i)5-s + 0.408i·6-s + (−0.948 − 0.315i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.0768 − 0.133i)10-s + (0.799 − 0.461i)11-s + (−0.249 − 0.144i)12-s + 1.40i·13-s + (0.528 − 0.469i)14-s + 0.125i·15-s + (−0.125 + 0.216i)16-s + (0.101 + 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.999 - 0.0155i$
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: \((2,\ 966,\ (\ :1/2),\ 0.999 - 0.0155i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41947 + 0.0110574i\)
\(L(\frac12)\) \(\approx\) \(1.41947 + 0.0110574i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.51 + 0.834i)T \)
23 \( 1 + (2.14 + 4.29i)T \)
good5 \( 1 + (0.242 - 0.420i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.65 + 1.53i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.07iT - 13T^{2} \)
17 \( 1 + (-0.418 - 0.725i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.41 + 5.91i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 - 4.62T + 29T^{2} \)
31 \( 1 + (-7.97 + 4.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.82 - 2.78i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.68iT - 41T^{2} \)
43 \( 1 + 5.27iT - 43T^{2} \)
47 \( 1 + (-3.68 - 2.12i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.31 - 4.22i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.15 - 1.82i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.90 + 8.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.54 + 2.04i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.32T + 71T^{2} \)
73 \( 1 + (-3.26 + 1.88i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.12 - 5.26i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.95T + 83T^{2} \)
89 \( 1 + (-5.22 + 9.04i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.53T + 97T^{2} \)
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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

−9.611465464036390511292535607318, −9.209315667310762130763056255890, −8.427042662551640247721336074712, −7.33973296128246660994737950805, −6.67739415262185407299384684636, −6.20826652757197366328709075059, −4.67822283210389536522276399493, −3.73560495070474650501582508977, −2.55626935972125348212093758475, −0.887819101999745879731593843392, 1.15213169687928439669154752539, 2.75568812568621342909918982641, 3.40953439061767916794716852728, 4.45285217159529016262218522420, 5.65829869782487334286529125059, 6.67711428978012280196298927348, 7.896160583938030698566225716164, 8.354506752461603662143489121966, 9.521031710498998194701511983493, 9.843370474608342180614121948077

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