Properties

Degree $2$
Conductor $1152$
Sign $0.959 + 0.282i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.909 + 0.909i)5-s + 0.654·7-s + (13.3 − 13.3i)11-s + (−8.32 + 8.32i)13-s + 3.93·17-s + (16.8 + 16.8i)19-s − 23.1·23-s − 23.3i·25-s + (35.6 − 35.6i)29-s + 45.5i·31-s + (0.595 + 0.595i)35-s + (−10.1 − 10.1i)37-s − 28.4i·41-s + (22.7 − 22.7i)43-s − 10.7i·47-s + ⋯
L(s)  = 1  + (0.181 + 0.181i)5-s + 0.0935·7-s + (1.21 − 1.21i)11-s + (−0.640 + 0.640i)13-s + 0.231·17-s + (0.889 + 0.889i)19-s − 1.00·23-s − 0.933i·25-s + (1.22 − 1.22i)29-s + 1.46i·31-s + (0.0170 + 0.0170i)35-s + (−0.274 − 0.274i)37-s − 0.694i·41-s + (0.528 − 0.528i)43-s − 0.229i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.959 + 0.282i$
Motivic weight: \(2\)
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ 0.959 + 0.282i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.152188835\)
\(L(\frac12)\) \(\approx\) \(2.152188835\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-0.909 - 0.909i)T + 25iT^{2} \)
7 \( 1 - 0.654T + 49T^{2} \)
11 \( 1 + (-13.3 + 13.3i)T - 121iT^{2} \)
13 \( 1 + (8.32 - 8.32i)T - 169iT^{2} \)
17 \( 1 - 3.93T + 289T^{2} \)
19 \( 1 + (-16.8 - 16.8i)T + 361iT^{2} \)
23 \( 1 + 23.1T + 529T^{2} \)
29 \( 1 + (-35.6 + 35.6i)T - 841iT^{2} \)
31 \( 1 - 45.5iT - 961T^{2} \)
37 \( 1 + (10.1 + 10.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 28.4iT - 1.68e3T^{2} \)
43 \( 1 + (-22.7 + 22.7i)T - 1.84e3iT^{2} \)
47 \( 1 + 10.7iT - 2.20e3T^{2} \)
53 \( 1 + (-41.5 - 41.5i)T + 2.80e3iT^{2} \)
59 \( 1 + (-21.0 + 21.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (-68.7 + 68.7i)T - 3.72e3iT^{2} \)
67 \( 1 + (-67.8 - 67.8i)T + 4.48e3iT^{2} \)
71 \( 1 - 33.3T + 5.04e3T^{2} \)
73 \( 1 - 18.6iT - 5.32e3T^{2} \)
79 \( 1 - 6.29iT - 6.24e3T^{2} \)
83 \( 1 + (-72.0 - 72.0i)T + 6.88e3iT^{2} \)
89 \( 1 - 10.6iT - 7.92e3T^{2} \)
97 \( 1 - 143.T + 9.40e3T^{2} \)
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

−9.672461569401080757928407598861, −8.681158961221507835566199647004, −8.077507342984178875448761761749, −6.93293885022585827487598551046, −6.25911280151017487462682187579, −5.42368475363009764966934423432, −4.21292965413262698394301701683, −3.40963412201781933671141841840, −2.13829315130620250707441961140, −0.835647616734344109998864347392, 1.01236126514455232198926141294, 2.20477640284435797901819954534, 3.43570861360936629331968189453, 4.55335868491476053465524911727, 5.24896707322265160618030971235, 6.37890918883922519192669181387, 7.18456394696865100254930102706, 7.895864965251895691539721132264, 9.004505267193249454321848372307, 9.666201886484629595359810688013

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