↑

Properties

Label 2-1134-63.41-c1-0-7
Degree $2$
Conductor $1134$
Sign $-0.305 - 0.952i$
Analytic cond. $9.05503$
Root an. cond. $3.00915$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.22 + 2.12i)5-s + (2.62 − 0.358i)7-s + 0.999i·8-s − 2.44i·10-s + (2.12 + 1.22i)13-s + (−2.09 + 1.62i)14-s + (−0.5 − 0.866i)16-s − 4.89·17-s + 2.44i·19-s + (1.22 + 2.12i)20-s + (5.19 + 3i)23-s + (−0.499 − 0.866i)25-s − 2.44·26-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.547 + 0.948i)5-s + (0.990 − 0.135i)7-s + 0.353i·8-s − 0.774i·10-s + (0.588 + 0.339i)13-s + (−0.558 + 0.433i)14-s + (−0.125 − 0.216i)16-s − 1.18·17-s + 0.561i·19-s + (0.273 + 0.474i)20-s + (1.08 + 0.625i)23-s + (−0.0999 − 0.173i)25-s − 0.480·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $-0.305 - 0.952i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ -0.305 - 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.073500296\)
\(L(\frac12)\) \(\approx\) \(1.073500296\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-2.62 + 0.358i)T \)
good5 \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.12 - 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.19 + 3i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (2.44 - 4.24i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.44 - 4.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + (6.12 - 10.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (10.6 - 6.12i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 + (-5 - 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.22 - 2.12i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-4.24 + 2.44i)T + (48.5 - 84.0i)T^{2} \)
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

−10.17259827353491561611659166445, −9.023198592971266553492940349064, −8.416155525103306853488052721658, −7.54548349218078713149815756938, −6.94168532191474624115050824296, −6.10993895565556801555079961973, −4.93340892897515556711113092601, −3.95458978231539740410451644877, −2.70302185799391594790098903518, −1.38737130034628442645727155079, 0.63274818063424262931442414244, 1.82595933138260372388140615981, 3.15631032968940887655490654387, 4.52200534331849200602781435255, 4.92118945940107980142329200165, 6.34246986771830949315336483078, 7.32864622538862597355887081396, 8.269299794568233338579957876959, 8.697013195404200271642984300081, 9.267433185839377639258761926026

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