↑

Properties

Label 2-966-161.68-c1-0-21
Degree $2$
Conductor $966$
Sign $0.843 + 0.537i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.347 − 0.601i)5-s + 0.999i·6-s + (2.64 − 0.158i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (0.347 + 0.601i)10-s + (0.872 − 0.503i)11-s + (−0.866 − 0.499i)12-s − 3.43i·13-s + (−1.18 + 2.36i)14-s − 0.694i·15-s + (−0.5 + 0.866i)16-s + (−3.19 − 5.53i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.155 − 0.268i)5-s + 0.408i·6-s + (0.998 − 0.0598i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.109 + 0.190i)10-s + (0.263 − 0.151i)11-s + (−0.249 − 0.144i)12-s − 0.953i·13-s + (−0.316 + 0.632i)14-s − 0.179i·15-s + (−0.125 + 0.216i)16-s + (−0.775 − 1.34i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.843 + 0.537i$
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: \((2,\ 966,\ (\ :1/2),\ 0.843 + 0.537i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62739 - 0.474300i\)
\(L(\frac12)\) \(\approx\) \(1.62739 - 0.474300i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.158i)T \)
23 \( 1 + (2.30 - 4.20i)T \)
good5 \( 1 + (-0.347 + 0.601i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.872 + 0.503i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.43iT - 13T^{2} \)
17 \( 1 + (3.19 + 5.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.60 + 4.50i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 2.86T + 29T^{2} \)
31 \( 1 + (6.88 - 3.97i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.39 - 1.38i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 5.92iT - 41T^{2} \)
43 \( 1 + 3.64iT - 43T^{2} \)
47 \( 1 + (-5.01 - 2.89i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.05 + 1.76i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.86 - 2.80i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.86 + 10.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-11.8 + 6.81i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 + (-2.25 + 1.30i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.80 - 4.50i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.72T + 83T^{2} \)
89 \( 1 + (7.07 - 12.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.480067404631326178029311006887, −9.120002245415449067178388075497, −8.216427200504249376318663842965, −7.43993669309650210319278175995, −6.89472357502474555689202406179, −5.44320683452137020558240437141, −5.02091280914838341150030031919, −3.62414589621674156808578178934, −2.23467255000268282902764119922, −0.898547476524837398894527253236, 1.63425509813131265269624170047, 2.38270891574269677919476257337, 3.90613692170523081449116450594, 4.37244909036781336891749162315, 5.73312301705038778679033975277, 6.85369475046573724730678413911, 7.924876183114351691912546325934, 8.497450694580636076010020208165, 9.291287759502160162680082930224, 10.11862193672694982875450054989

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