↑

Properties

Label 2-1176-21.20-c1-0-24
Degree $2$
Conductor $1176$
Sign $0.506 + 0.862i$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
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  + (1.72 − 0.188i)3-s − 4.01·5-s + (2.92 − 0.650i)9-s + 3.77i·11-s − 4.48i·13-s + (−6.91 + 0.758i)15-s + 4.11·17-s − 6.57i·19-s − 5.01i·23-s + 11.1·25-s + (4.91 − 1.67i)27-s − 1.23i·29-s − 0.934i·31-s + (0.712 + 6.49i)33-s + 2.02·37-s + ⋯
L(s)  = 1  + (0.994 − 0.109i)3-s − 1.79·5-s + (0.976 − 0.216i)9-s + 1.13i·11-s − 1.24i·13-s + (−1.78 + 0.195i)15-s + 0.999·17-s − 1.50i·19-s − 1.04i·23-s + 2.22·25-s + (0.946 − 0.322i)27-s − 0.228i·29-s − 0.167i·31-s + (0.124 + 1.13i)33-s + 0.332·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ 0.506 + 0.862i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.595805889\)
\(L(\frac12)\) \(\approx\) \(1.595805889\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.72 + 0.188i)T \)
7 \( 1 \)
good5 \( 1 + 4.01T + 5T^{2} \)
11 \( 1 - 3.77iT - 11T^{2} \)
13 \( 1 + 4.48iT - 13T^{2} \)
17 \( 1 - 4.11T + 17T^{2} \)
19 \( 1 + 6.57iT - 19T^{2} \)
23 \( 1 + 5.01iT - 23T^{2} \)
29 \( 1 + 1.23iT - 29T^{2} \)
31 \( 1 + 0.934iT - 31T^{2} \)
37 \( 1 - 2.02T + 37T^{2} \)
41 \( 1 - 2.21T + 41T^{2} \)
43 \( 1 + 3.14T + 43T^{2} \)
47 \( 1 - 5.85T + 47T^{2} \)
53 \( 1 + 4.56iT - 53T^{2} \)
59 \( 1 + 1.96T + 59T^{2} \)
61 \( 1 + 9.24iT - 61T^{2} \)
67 \( 1 - 7.76T + 67T^{2} \)
71 \( 1 - 7.97iT - 71T^{2} \)
73 \( 1 + 14.8iT - 73T^{2} \)
79 \( 1 - 3.70T + 79T^{2} \)
83 \( 1 + 9.15T + 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 - 7.06iT - 97T^{2} \)
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

−9.548290440852545968775876487108, −8.584351168551879279819102432799, −7.940161451625112332994070395436, −7.43944797689701717047279253776, −6.71982094626067800167110259207, −5.01130026217328111368455478864, −4.27989401602941617689611655646, −3.38479773528111422061443005617, −2.55335846439121026076186188528, −0.70018735682172447747636748699, 1.32382949346557656158618974032, 3.03303527697273413761503404280, 3.74574700132237882368766843813, 4.27096991539990531828399939165, 5.63458300774574447808397521771, 6.94698905309635410405210441988, 7.66501198478391187433666037098, 8.221747220038447585542836962850, 8.859490116050501633555585764996, 9.775490328670132973825026044282

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