↑

Properties

Label 2-1176-3.2-c2-0-29
Degree $2$
Conductor $1176$
Sign $0.639 - 0.769i$
Analytic cond. $32.0436$
Root an. cond. $5.66071$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.30 − 1.91i)3-s + 7.47i·5-s + (1.64 + 8.84i)9-s − 6.94i·11-s + 25.0·13-s + (14.3 − 17.2i)15-s − 0.410i·17-s − 20.2·19-s − 27.3i·23-s − 30.9·25-s + (13.1 − 23.5i)27-s + 11.9i·29-s + 41.5·31-s + (−13.3 + 16.0i)33-s + 32.9·37-s + ⋯
L(s)  = 1  + (−0.769 − 0.639i)3-s + 1.49i·5-s + (0.182 + 0.983i)9-s − 0.631i·11-s + 1.92·13-s + (0.956 − 1.15i)15-s − 0.0241i·17-s − 1.06·19-s − 1.18i·23-s − 1.23·25-s + (0.487 − 0.872i)27-s + 0.411i·29-s + 1.34·31-s + (−0.403 + 0.485i)33-s + 0.889·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.639 - 0.769i$
Analytic conductor: \(32.0436\)
Root analytic conductor: \(5.66071\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1),\ 0.639 - 0.769i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.447548013\)
\(L(\frac12)\) \(\approx\) \(1.447548013\)
\(L(2)\) 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 \( 1 + (2.30 + 1.91i)T \)
7 \( 1 \)
good5 \( 1 - 7.47iT - 25T^{2} \)
11 \( 1 + 6.94iT - 121T^{2} \)
13 \( 1 - 25.0T + 169T^{2} \)
17 \( 1 + 0.410iT - 289T^{2} \)
19 \( 1 + 20.2T + 361T^{2} \)
23 \( 1 + 27.3iT - 529T^{2} \)
29 \( 1 - 11.9iT - 841T^{2} \)
31 \( 1 - 41.5T + 961T^{2} \)
37 \( 1 - 32.9T + 1.36e3T^{2} \)
41 \( 1 - 45.7iT - 1.68e3T^{2} \)
43 \( 1 + 66.3T + 1.84e3T^{2} \)
47 \( 1 - 51.4iT - 2.20e3T^{2} \)
53 \( 1 + 40.4iT - 2.80e3T^{2} \)
59 \( 1 - 54.9iT - 3.48e3T^{2} \)
61 \( 1 - 114.T + 3.72e3T^{2} \)
67 \( 1 + 21.1T + 4.48e3T^{2} \)
71 \( 1 - 81.6iT - 5.04e3T^{2} \)
73 \( 1 - 23.9T + 5.32e3T^{2} \)
79 \( 1 - 31.7T + 6.24e3T^{2} \)
83 \( 1 + 11.6iT - 6.88e3T^{2} \)
89 \( 1 + 141. iT - 7.92e3T^{2} \)
97 \( 1 + 45.9T + 9.40e3T^{2} \)
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

−10.10556542564956260649129952971, −8.551688298393371550441218807641, −8.094945786614974722212891180767, −6.85553267289852334679450731019, −6.38011460508503663717562973656, −5.94273884431278855577240235409, −4.51312787769179388510508433851, −3.37219525177763716256311439801, −2.38125683237709559644948449425, −0.990822749998795765514215025903, 0.62486318723333723020632551424, 1.65895167757536804394564538446, 3.66291939242348385071252431402, 4.32721432739046003291672739591, 5.15708731998664890345507749177, 5.92780869319111066304587586419, 6.71593789951848280643883871374, 8.156051046580915271339775417260, 8.691912690471510086605046948115, 9.477713591883511422548608439351

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