Properties

Label 2-1152-8.5-c3-0-28
Degree $2$
Conductor $1152$
Sign $1$
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  + 17.8i·5-s − 22.6·7-s − 44.2i·11-s + 17.8i·13-s − 70·17-s − 82.2i·19-s + 158.·23-s − 195.·25-s + 125. i·29-s − 404. i·35-s − 375. i·37-s + 182·41-s − 132. i·43-s − 316.·47-s + 169.·49-s + ⋯
L(s)  = 1  + 1.59i·5-s − 1.22·7-s − 1.21i·11-s + 0.381i·13-s − 0.998·17-s − 0.992i·19-s + 1.43·23-s − 1.56·25-s + 0.801i·29-s − 1.95i·35-s − 1.66i·37-s + 0.693·41-s − 0.471i·43-s − 0.983·47-s + 0.492·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $1$
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: \((2,\ 1152,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.314160053\)
\(L(\frac12)\) \(\approx\) \(1.314160053\)
\(L(\frac{5}{2})\) 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 \( 1 - 17.8iT - 125T^{2} \)
7 \( 1 + 22.6T + 343T^{2} \)
11 \( 1 + 44.2iT - 1.33e3T^{2} \)
13 \( 1 - 17.8iT - 2.19e3T^{2} \)
17 \( 1 + 70T + 4.91e3T^{2} \)
19 \( 1 + 82.2iT - 6.85e3T^{2} \)
23 \( 1 - 158.T + 1.21e4T^{2} \)
29 \( 1 - 125. iT - 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 + 375. iT - 5.06e4T^{2} \)
41 \( 1 - 182T + 6.89e4T^{2} \)
43 \( 1 + 132. iT - 7.95e4T^{2} \)
47 \( 1 + 316.T + 1.03e5T^{2} \)
53 \( 1 + 125. iT - 1.48e5T^{2} \)
59 \( 1 - 82.2iT - 2.05e5T^{2} \)
61 \( 1 - 232. iT - 2.26e5T^{2} \)
67 \( 1 - 221. iT - 3.00e5T^{2} \)
71 \( 1 - 113.T + 3.57e5T^{2} \)
73 \( 1 - 910T + 3.89e5T^{2} \)
79 \( 1 - 678.T + 4.93e5T^{2} \)
83 \( 1 - 714. iT - 5.71e5T^{2} \)
89 \( 1 - 546T + 7.04e5T^{2} \)
97 \( 1 + 490T + 9.12e5T^{2} \)
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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

−9.319191630723803869816689832846, −8.867088036165347399582991566143, −7.50994398730473879851690874466, −6.63726675884539034109106469708, −6.50934994326694564621360151637, −5.31008946437703544524167275836, −3.83455927725786060119868363253, −3.10430814164095505884132287060, −2.42838453537155661987102857042, −0.47231757526067068572610863332, 0.70515643916818248835163208312, 1.88337866719055649217695386256, 3.22418600486866050122847410967, 4.40899517226769976302220527345, 4.96347475837193427802990757268, 6.05830774221144873085812928053, 6.86214296114207818759319386300, 7.921430874119922447323193223720, 8.673684342845594809792989673379, 9.636510526807099895753218902705

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