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Properties

Label 2-1152-8.5-c3-0-4
Degree $2$
Conductor $1152$
Sign $-0.707 - 0.707i$
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  + 4i·5-s − 92i·13-s − 94·17-s + 109·25-s + 284i·29-s + 396i·37-s + 230·41-s − 343·49-s − 572i·53-s + 468i·61-s + 368·65-s − 1.09e3·73-s − 376i·85-s − 1.67e3·89-s − 594·97-s + ⋯
L(s)  = 1  + 0.357i·5-s − 1.96i·13-s − 1.34·17-s + 0.871·25-s + 1.81i·29-s + 1.75i·37-s + 0.876·41-s − 49-s − 1.48i·53-s + 0.982i·61-s + 0.702·65-s − 1.76·73-s − 0.479i·85-s − 1.98·89-s − 0.621·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6566325527\)
\(L(\frac12)\) \(\approx\) \(0.6566325527\)
\(L(\frac{5}{2})\) 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 \( 1 - 4iT - 125T^{2} \)
7 \( 1 + 343T^{2} \)
11 \( 1 - 1.33e3T^{2} \)
13 \( 1 + 92iT - 2.19e3T^{2} \)
17 \( 1 + 94T + 4.91e3T^{2} \)
19 \( 1 - 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 284iT - 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 - 396iT - 5.06e4T^{2} \)
41 \( 1 - 230T + 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 572iT - 1.48e5T^{2} \)
59 \( 1 - 2.05e5T^{2} \)
61 \( 1 - 468iT - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 1.09e3T + 3.89e5T^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 + 1.67e3T + 7.04e5T^{2} \)
97 \( 1 + 594T + 9.12e5T^{2} \)
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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

−9.846721667013930985143600456161, −8.770756777207221019181577247295, −8.186543443507164219810449822741, −7.18335257154799982672761882074, −6.47652226528099738896484356665, −5.44298612168964414983687634898, −4.67019392969899697858875913045, −3.34910024260917695180001711153, −2.66915897433095959557832077903, −1.18098686291335506144820966956, 0.16134030992495487365135090624, 1.65348811902961442924815887329, 2.57686930541565764485128091400, 4.19187371631371860693197531896, 4.47175676802858234722726450670, 5.81449674838299737723670751083, 6.63330759005090478821176952031, 7.36071712365127121825596315504, 8.475707962570768734630610448643, 9.145224804552795251189529639975

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