Properties

Label 2-288-8.5-c5-0-12
Degree $2$
Conductor $288$
Sign $0.843 + 0.537i$
Analytic cond. $46.1905$
Root an. cond. $6.79636$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.38i·5-s − 160.·7-s − 129. i·11-s + 759. i·13-s − 323.·17-s − 198. i·19-s − 1.19e3·23-s + 3.12e3·25-s − 5.98e3i·29-s + 4.87e3·31-s + 222. i·35-s − 3.69e3i·37-s + 1.04e4·41-s + 9.87e3i·43-s − 6.29e3·47-s + ⋯
L(s)  = 1  − 0.0247i·5-s − 1.23·7-s − 0.321i·11-s + 1.24i·13-s − 0.271·17-s − 0.126i·19-s − 0.470·23-s + 0.999·25-s − 1.32i·29-s + 0.910·31-s + 0.0307i·35-s − 0.444i·37-s + 0.969·41-s + 0.814i·43-s − 0.415·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.843 + 0.537i$
Analytic conductor: \(46.1905\)
Root analytic conductor: \(6.79636\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :5/2),\ 0.843 + 0.537i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.423873356\)
\(L(\frac12)\) \(\approx\) \(1.423873356\)
\(L(\frac{7}{2})\) 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 \( 1 + 1.38iT - 3.12e3T^{2} \)
7 \( 1 + 160.T + 1.68e4T^{2} \)
11 \( 1 + 129. iT - 1.61e5T^{2} \)
13 \( 1 - 759. iT - 3.71e5T^{2} \)
17 \( 1 + 323.T + 1.41e6T^{2} \)
19 \( 1 + 198. iT - 2.47e6T^{2} \)
23 \( 1 + 1.19e3T + 6.43e6T^{2} \)
29 \( 1 + 5.98e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.87e3T + 2.86e7T^{2} \)
37 \( 1 + 3.69e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.04e4T + 1.15e8T^{2} \)
43 \( 1 - 9.87e3iT - 1.47e8T^{2} \)
47 \( 1 + 6.29e3T + 2.29e8T^{2} \)
53 \( 1 + 2.17e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.35e4iT - 7.14e8T^{2} \)
61 \( 1 + 4.85e4iT - 8.44e8T^{2} \)
67 \( 1 + 3.31e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.94e4T + 1.80e9T^{2} \)
73 \( 1 - 5.12e4T + 2.07e9T^{2} \)
79 \( 1 - 7.37e4T + 3.07e9T^{2} \)
83 \( 1 + 6.16e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.06e5T + 5.58e9T^{2} \)
97 \( 1 - 1.25e4T + 8.58e9T^{2} \)
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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

−10.90404054043884902667984011190, −9.755013691841420947030381622494, −9.188736839143973598558126814003, −8.022183387557427283346709360864, −6.71126608463499125381783007705, −6.18184632296640829864592583885, −4.65525050074578693126863890416, −3.52699034977226172506393414421, −2.28081684856554294278192896635, −0.54543966784961438716635172697, 0.811138877794429212437438204470, 2.63450772598622689659552819522, 3.59512446852714993596139320887, 5.03430027716428851880899563511, 6.17421100959674518707073979630, 7.04807935144683120879317297867, 8.191875417616264626066574729470, 9.251547355972995621733537971656, 10.14270081095083580207701261713, 10.84020334647214684597804972588

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