↑

Properties

Label 2-966-161.45-c1-0-18
Degree $2$
Conductor $966$
Sign $0.998 + 0.0578i$
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 + (−0.139 − 0.240i)5-s − 0.999i·6-s + (2.42 − 1.05i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.139 − 0.240i)10-s + (1.25 + 0.726i)11-s + (0.866 − 0.499i)12-s − 5.02i·13-s + (2.12 + 1.57i)14-s + 0.278i·15-s + (−0.5 − 0.866i)16-s + (−2.48 + 4.31i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.0622 − 0.107i)5-s − 0.408i·6-s + (0.917 − 0.397i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.0439 − 0.0762i)10-s + (0.379 + 0.219i)11-s + (0.249 − 0.144i)12-s − 1.39i·13-s + (0.567 + 0.421i)14-s + 0.0718i·15-s + (−0.125 − 0.216i)16-s + (−0.603 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.998 + 0.0578i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.998 + 0.0578i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69257 - 0.0490055i\)
\(L(\frac12)\) \(\approx\) \(1.69257 - 0.0490055i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.42 + 1.05i)T \)
23 \( 1 + (-2.68 + 3.97i)T \)
good5 \( 1 + (0.139 + 0.240i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.25 - 0.726i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.02iT - 13T^{2} \)
17 \( 1 + (2.48 - 4.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.02 + 1.78i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 5.24T + 29T^{2} \)
31 \( 1 + (-0.516 - 0.298i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.31 + 1.33i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.01iT - 41T^{2} \)
43 \( 1 - 6.84iT - 43T^{2} \)
47 \( 1 + (-8.05 + 4.65i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.65 - 4.99i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.40 - 1.38i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.995 - 1.72i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.81 - 3.35i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.995T + 71T^{2} \)
73 \( 1 + (13.8 + 7.97i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.24 + 1.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 + (-3.83 - 6.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 4.93T + 97T^{2} \)
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

−10.34282157966430039886695110898, −8.772529475573559767542757608168, −8.274883326396336655112101910842, −7.38691224867446932205834810885, −6.58192294687455437516373606440, −5.72709652046340847920179563299, −4.77692123462913188663588248035, −4.11941779063941802543394114199, −2.56162129440949025587173329325, −0.891693080468038501806957498392, 1.29851205681642380140396029923, 2.52333876135118454691986337575, 3.88400232257893272678994108761, 4.70362330199796541534190130760, 5.42492310186356497838375133678, 6.51106222190811082492279633253, 7.35074744364931889501733266394, 8.743098195943475414755429330244, 9.189534912019513040946341860206, 10.17679177909774315824445244935

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