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Properties

Label 2-1152-48.11-c3-0-11
Degree $2$
Conductor $1152$
Sign $-0.129 - 0.991i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.31 + 6.31i)5-s + 16.2·7-s + (−48.1 − 48.1i)11-s + (8.61 − 8.61i)13-s + 53.2i·17-s + (55.5 + 55.5i)19-s − 66.9i·23-s + 45.2i·25-s + (−126. − 126. i)29-s − 121. i·31-s + (−102. + 102. i)35-s + (250. + 250. i)37-s + 402.·41-s + (−187. + 187. i)43-s − 96.1·47-s + ⋯
L(s)  = 1  + (−0.564 + 0.564i)5-s + 0.878·7-s + (−1.31 − 1.31i)11-s + (0.183 − 0.183i)13-s + 0.759i·17-s + (0.670 + 0.670i)19-s − 0.607i·23-s + 0.361i·25-s + (−0.809 − 0.809i)29-s − 0.701i·31-s + (−0.496 + 0.496i)35-s + (1.11 + 1.11i)37-s + 1.53·41-s + (−0.664 + 0.664i)43-s − 0.298·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.129 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.129 - 0.991i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ -0.129 - 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.251138653\)
\(L(\frac12)\) \(\approx\) \(1.251138653\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.31 - 6.31i)T - 125iT^{2} \)
7 \( 1 - 16.2T + 343T^{2} \)
11 \( 1 + (48.1 + 48.1i)T + 1.33e3iT^{2} \)
13 \( 1 + (-8.61 + 8.61i)T - 2.19e3iT^{2} \)
17 \( 1 - 53.2iT - 4.91e3T^{2} \)
19 \( 1 + (-55.5 - 55.5i)T + 6.85e3iT^{2} \)
23 \( 1 + 66.9iT - 1.21e4T^{2} \)
29 \( 1 + (126. + 126. i)T + 2.43e4iT^{2} \)
31 \( 1 + 121. iT - 2.97e4T^{2} \)
37 \( 1 + (-250. - 250. i)T + 5.06e4iT^{2} \)
41 \( 1 - 402.T + 6.89e4T^{2} \)
43 \( 1 + (187. - 187. i)T - 7.95e4iT^{2} \)
47 \( 1 + 96.1T + 1.03e5T^{2} \)
53 \( 1 + (-90.3 + 90.3i)T - 1.48e5iT^{2} \)
59 \( 1 + (-488. - 488. i)T + 2.05e5iT^{2} \)
61 \( 1 + (378. - 378. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-223. - 223. i)T + 3.00e5iT^{2} \)
71 \( 1 + 231. iT - 3.57e5T^{2} \)
73 \( 1 - 265. iT - 3.89e5T^{2} \)
79 \( 1 + 604. iT - 4.93e5T^{2} \)
83 \( 1 + (351. - 351. i)T - 5.71e5iT^{2} \)
89 \( 1 + 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + 1.85e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.781956574279964668579469978475, −8.493584183863140857443723245052, −7.994280007478341091587358243388, −7.47370254380030207570870994489, −6.10570500912640084037322201211, −5.53950844395978296979761854990, −4.38824510512910179518745252128, −3.40096765263864802616771869495, −2.49422831262116981599863667016, −1.02506407368512083346969954706, 0.34280835486553371803721138485, 1.68998663650876883418043947867, 2.76962401825645084705539628312, 4.15861889758684257396540120507, 4.93567613161123572334099072510, 5.43850750685278096181409578729, 7.03186235378419473307055446007, 7.58450859317612261378565093384, 8.238726051425766767701353672020, 9.223771780718425035854136919889

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