Properties

Label 2-1152-72.29-c0-0-0
Degree $2$
Conductor $1152$
Sign $-0.342 - 0.939i$
Analytic cond. $0.574922$
Root an. cond. $0.758236$
Motivic weight $0$
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)3-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 1.73i·17-s + 1.73i·19-s + (0.5 − 0.866i)25-s + 0.999·27-s + (−0.499 − 0.866i)33-s + (−1.5 + 0.866i)41-s + (−1.5 − 0.866i)43-s + (0.5 + 0.866i)49-s + (−1.49 − 0.866i)51-s + (−1.49 − 0.866i)57-s + (0.5 + 0.866i)59-s + (1.5 − 0.866i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 1.73i·17-s + 1.73i·19-s + (0.5 − 0.866i)25-s + 0.999·27-s + (−0.499 − 0.866i)33-s + (−1.5 + 0.866i)41-s + (−1.5 − 0.866i)43-s + (0.5 + 0.866i)49-s + (−1.49 − 0.866i)51-s + (−1.49 − 0.866i)57-s + (0.5 + 0.866i)59-s + (1.5 − 0.866i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.342 - 0.939i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :0),\ -0.342 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7444951891\)
\(L(\frac12)\) \(\approx\) \(0.7444951891\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.73iT - T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−10.30401613582429353912948649270, −9.729923335212448247062228102154, −8.561260906443996324119220963297, −8.005446621875925014411523275044, −6.70989330109287694797106737161, −5.97533084442321874325234846217, −5.09349761546224666032753480224, −4.19236168335300638227054785059, −3.38229133065129211881688170383, −1.82502763960839879152377636775, 0.71063606592494478887874750811, 2.35311361942562411350292863515, 3.26355754649610131439298947658, 5.06196629728256267700134865641, 5.26936110194668502827172135590, 6.70528334974994246019662780858, 7.00871047801996196995493506003, 8.059808779249993604386062059198, 8.807684935307733772315429504973, 9.713698889122175401365338352956

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