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Properties

Label 2-1152-48.35-c3-0-28
Degree $2$
Conductor $1152$
Sign $0.511 + 0.859i$
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  + (−2.40 − 2.40i)5-s − 11.7·7-s + (34.7 − 34.7i)11-s + (3.17 + 3.17i)13-s + 98.0i·17-s + (15.9 − 15.9i)19-s + 69.6i·23-s − 113. i·25-s + (15.9 − 15.9i)29-s + 121. i·31-s + (28.2 + 28.2i)35-s + (37.0 − 37.0i)37-s + 59.3·41-s + (241. + 241. i)43-s − 395.·47-s + ⋯
L(s)  = 1  + (−0.215 − 0.215i)5-s − 0.632·7-s + (0.952 − 0.952i)11-s + (0.0678 + 0.0678i)13-s + 1.39i·17-s + (0.192 − 0.192i)19-s + 0.631i·23-s − 0.907i·25-s + (0.102 − 0.102i)29-s + 0.702i·31-s + (0.136 + 0.136i)35-s + (0.164 − 0.164i)37-s + 0.225·41-s + (0.857 + 0.857i)43-s − 1.22·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.511 + 0.859i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ 0.511 + 0.859i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.666606192\)
\(L(\frac12)\) \(\approx\) \(1.666606192\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.40 + 2.40i)T + 125iT^{2} \)
7 \( 1 + 11.7T + 343T^{2} \)
11 \( 1 + (-34.7 + 34.7i)T - 1.33e3iT^{2} \)
13 \( 1 + (-3.17 - 3.17i)T + 2.19e3iT^{2} \)
17 \( 1 - 98.0iT - 4.91e3T^{2} \)
19 \( 1 + (-15.9 + 15.9i)T - 6.85e3iT^{2} \)
23 \( 1 - 69.6iT - 1.21e4T^{2} \)
29 \( 1 + (-15.9 + 15.9i)T - 2.43e4iT^{2} \)
31 \( 1 - 121. iT - 2.97e4T^{2} \)
37 \( 1 + (-37.0 + 37.0i)T - 5.06e4iT^{2} \)
41 \( 1 - 59.3T + 6.89e4T^{2} \)
43 \( 1 + (-241. - 241. i)T + 7.95e4iT^{2} \)
47 \( 1 + 395.T + 1.03e5T^{2} \)
53 \( 1 + (458. + 458. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-257. + 257. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-373. - 373. i)T + 2.26e5iT^{2} \)
67 \( 1 + (-648. + 648. i)T - 3.00e5iT^{2} \)
71 \( 1 + 787. iT - 3.57e5T^{2} \)
73 \( 1 + 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 - 382. iT - 4.93e5T^{2} \)
83 \( 1 + (-491. - 491. i)T + 5.71e5iT^{2} \)
89 \( 1 - 624.T + 7.04e5T^{2} \)
97 \( 1 - 1.66e3T + 9.12e5T^{2} \)
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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

−9.259559468132732558673882384221, −8.481981003032991446273402736305, −7.79836902686493757384865804877, −6.43608968638593690850657789493, −6.24499076615070473390843691941, −4.96390612773082137297579002139, −3.85008134389582688288455797456, −3.22745387070012195461913377233, −1.71355378133012100231113666368, −0.51354832171429263656689888064, 0.901357983599850634782591295731, 2.30125449753163202496841436708, 3.36183800590334729831379966763, 4.29102099188236775321396485536, 5.25564917404373904014126345223, 6.39468699118576013269020720733, 7.03399205042372878540605585880, 7.74571526297211461969266425019, 8.952522536494690889773242951279, 9.549201201555274758328866055125

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