Properties

Label 2-1152-1.1-c3-0-0
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 yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 15.8·5-s − 17.8·7-s − 52.9·11-s + 8.43·13-s − 129.·17-s + 50.4·19-s − 128.·23-s + 126.·25-s − 111.·29-s − 302.·31-s + 283.·35-s − 182.·37-s + 94.5·41-s + 184.·43-s − 296.·47-s − 24.1·49-s + 102.·53-s + 839.·55-s + 93.3·59-s − 338.·61-s − 133.·65-s + 489.·67-s + 86.9·71-s − 154.·73-s + 945.·77-s − 449.·79-s − 383.·83-s + ⋯
L(s)  = 1  − 1.41·5-s − 0.964·7-s − 1.45·11-s + 0.179·13-s − 1.84·17-s + 0.609·19-s − 1.16·23-s + 1.01·25-s − 0.712·29-s − 1.75·31-s + 1.36·35-s − 0.813·37-s + 0.360·41-s + 0.654·43-s − 0.921·47-s − 0.0704·49-s + 0.266·53-s + 2.05·55-s + 0.206·59-s − 0.711·61-s − 0.255·65-s + 0.891·67-s + 0.145·71-s − 0.247·73-s + 1.39·77-s − 0.640·79-s − 0.507·83-s + ⋯

Functional equation

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

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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.07445942389\)
\(L(\frac12)\) \(\approx\) \(0.07445942389\)
\(L(\frac{5}{2})\) 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 \( 1 + 15.8T + 125T^{2} \)
7 \( 1 + 17.8T + 343T^{2} \)
11 \( 1 + 52.9T + 1.33e3T^{2} \)
13 \( 1 - 8.43T + 2.19e3T^{2} \)
17 \( 1 + 129.T + 4.91e3T^{2} \)
19 \( 1 - 50.4T + 6.85e3T^{2} \)
23 \( 1 + 128.T + 1.21e4T^{2} \)
29 \( 1 + 111.T + 2.43e4T^{2} \)
31 \( 1 + 302.T + 2.97e4T^{2} \)
37 \( 1 + 182.T + 5.06e4T^{2} \)
41 \( 1 - 94.5T + 6.89e4T^{2} \)
43 \( 1 - 184.T + 7.95e4T^{2} \)
47 \( 1 + 296.T + 1.03e5T^{2} \)
53 \( 1 - 102.T + 1.48e5T^{2} \)
59 \( 1 - 93.3T + 2.05e5T^{2} \)
61 \( 1 + 338.T + 2.26e5T^{2} \)
67 \( 1 - 489.T + 3.00e5T^{2} \)
71 \( 1 - 86.9T + 3.57e5T^{2} \)
73 \( 1 + 154.T + 3.89e5T^{2} \)
79 \( 1 + 449.T + 4.93e5T^{2} \)
83 \( 1 + 383.T + 5.71e5T^{2} \)
89 \( 1 - 517.T + 7.04e5T^{2} \)
97 \( 1 - 1.73e3T + 9.12e5T^{2} \)
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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

−9.341455707355152474695867907517, −8.541156139187740521843782455698, −7.68562882331485092625582246454, −7.14515127532084360772116167814, −6.11056848394983146397834573417, −5.06168588086526375398345085785, −4.03301599112541618905661674764, −3.32602251315365970348583247998, −2.19384496334053942808076286460, −0.13040086961211512843735137077, 0.13040086961211512843735137077, 2.19384496334053942808076286460, 3.32602251315365970348583247998, 4.03301599112541618905661674764, 5.06168588086526375398345085785, 6.11056848394983146397834573417, 7.14515127532084360772116167814, 7.68562882331485092625582246454, 8.541156139187740521843782455698, 9.341455707355152474695867907517

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