↑

Properties

Label 2-1134-63.16-c1-0-11
Degree $2$
Conductor $1134$
Sign $0.975 - 0.220i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 5-s + (2 + 1.73i)7-s + 0.999·8-s + (−0.5 − 0.866i)10-s + 5·11-s + (0.499 − 2.59i)14-s + (−0.5 − 0.866i)16-s + (2 + 3.46i)17-s + (−4 + 6.92i)19-s + (−0.499 + 0.866i)20-s + (−2.5 − 4.33i)22-s − 4·23-s − 4·25-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 0.447·5-s + (0.755 + 0.654i)7-s + 0.353·8-s + (−0.158 − 0.273i)10-s + 1.50·11-s + (0.133 − 0.694i)14-s + (−0.125 − 0.216i)16-s + (0.485 + 0.840i)17-s + (−0.917 + 1.58i)19-s + (−0.111 + 0.193i)20-s + (−0.533 − 0.923i)22-s − 0.834·23-s − 0.800·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1134\)    =    \(2 \cdot 3^{4} \cdot 7\)
Sign: $0.975 - 0.220i$
Analytic conductor: \(9.05503\)
Root analytic conductor: \(3.00915\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1134} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1134,\ (\ :1/2),\ 0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.612144843\)
\(L(\frac12)\) \(\approx\) \(1.612144843\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2 - 1.73i)T \)
good5 \( 1 - T + 5T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 2.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.5 + 2.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.5 + 6.06i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.5 - 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.885091742322668318154652261159, −9.077556515309821629133765072707, −8.353306665935303634452004410265, −7.69186550839023917050464081808, −6.24613804831363455531798114053, −5.80513449795066747172927171629, −4.36105895330775565128051570096, −3.68862911514877503112798619708, −2.13762194134070572957538737961, −1.47261802301261065866000371488, 0.889780786249069642587307314762, 2.13974118550436682004688967592, 3.84287671461810534205376915146, 4.67272674170603994530873488218, 5.62675121393736725652141129291, 6.69597695111171852071264192433, 7.12022006626006840660572310824, 8.206947165046150323438989344873, 8.917908129767219229642457217633, 9.668074273613551459688038262444

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