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Properties

Label 2-1152-8.5-c3-0-56
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  + 18.4i·5-s + 22.4·7-s − 53.6i·11-s − 7.15i·13-s − 39.6·17-s − 125. i·19-s − 99.1·23-s − 214.·25-s + 205. i·29-s − 147.·31-s + 413. i·35-s + 125. i·37-s − 506.·41-s − 413. i·43-s − 313.·47-s + ⋯
L(s)  = 1  + 1.64i·5-s + 1.21·7-s − 1.47i·11-s − 0.152i·13-s − 0.566·17-s − 1.51i·19-s − 0.898·23-s − 1.71·25-s + 1.31i·29-s − 0.854·31-s + 1.99i·35-s + 0.558i·37-s − 1.92·41-s − 1.46i·43-s − 0.974·47-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

\(L(2)\) \(\approx\) \(0.3017880815\)
\(L(\frac12)\) \(\approx\) \(0.3017880815\)
\(L(\frac{5}{2})\) 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 \( 1 - 18.4iT - 125T^{2} \)
7 \( 1 - 22.4T + 343T^{2} \)
11 \( 1 + 53.6iT - 1.33e3T^{2} \)
13 \( 1 + 7.15iT - 2.19e3T^{2} \)
17 \( 1 + 39.6T + 4.91e3T^{2} \)
19 \( 1 + 125. iT - 6.85e3T^{2} \)
23 \( 1 + 99.1T + 1.21e4T^{2} \)
29 \( 1 - 205. iT - 2.43e4T^{2} \)
31 \( 1 + 147.T + 2.97e4T^{2} \)
37 \( 1 - 125. iT - 5.06e4T^{2} \)
41 \( 1 + 506.T + 6.89e4T^{2} \)
43 \( 1 + 413. iT - 7.95e4T^{2} \)
47 \( 1 + 313.T + 1.03e5T^{2} \)
53 \( 1 - 44.3iT - 1.48e5T^{2} \)
59 \( 1 - 324iT - 2.05e5T^{2} \)
61 \( 1 + 324iT - 2.26e5T^{2} \)
67 \( 1 - 464. iT - 3.00e5T^{2} \)
71 \( 1 + 1.05e3T + 3.57e5T^{2} \)
73 \( 1 + 1.02e3T + 3.89e5T^{2} \)
79 \( 1 - 602.T + 4.93e5T^{2} \)
83 \( 1 + 15.8iT - 5.71e5T^{2} \)
89 \( 1 - 381.T + 7.04e5T^{2} \)
97 \( 1 - 659.T + 9.12e5T^{2} \)
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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

−8.909064102704463935953466915864, −8.345302907233649182043329903042, −7.31997514318510853229009342440, −6.73660119963369999253679671443, −5.79258111050766000520836999305, −4.85579691800265619490730407924, −3.58050804708095163974975248100, −2.84166766365816001793022049532, −1.73083344713812760762707842141, −0.06550150401118963810054651172, 1.55926159552243399233309041027, 1.89614750482167420165937717160, 3.98968452482039685194856849941, 4.61387043936934815483749938222, 5.22303203853621308770461437635, 6.24255281704662336077519130889, 7.64382163412879804639786032265, 8.035963104480302838212414774628, 8.846375976651408414473349532229, 9.672359576990741633165305511414

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