Properties

Label 2-72e2-24.11-c1-0-66
Degree $2$
Conductor $5184$
Sign $0.258 + 0.965i$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.206·5-s + 3.03i·7-s + 5.02i·11-s − 4.03i·13-s − 4.81i·17-s − 1.03·19-s − 5.43·23-s − 4.95·25-s − 2.90·29-s − 1.23i·31-s − 0.629i·35-s − 8.77i·37-s + 0.979i·41-s + 3.18·43-s + 3.26·47-s + ⋯
L(s)  = 1  − 0.0925·5-s + 1.14i·7-s + 1.51i·11-s − 1.12i·13-s − 1.16i·17-s − 0.238·19-s − 1.13·23-s − 0.991·25-s − 0.539·29-s − 0.222i·31-s − 0.106i·35-s − 1.44i·37-s + 0.152i·41-s + 0.485·43-s + 0.476·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.258 + 0.965i$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ 0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.019615846\)
\(L(\frac12)\) \(\approx\) \(1.019615846\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.206T + 5T^{2} \)
7 \( 1 - 3.03iT - 7T^{2} \)
11 \( 1 - 5.02iT - 11T^{2} \)
13 \( 1 + 4.03iT - 13T^{2} \)
17 \( 1 + 4.81iT - 17T^{2} \)
19 \( 1 + 1.03T + 19T^{2} \)
23 \( 1 + 5.43T + 23T^{2} \)
29 \( 1 + 2.90T + 29T^{2} \)
31 \( 1 + 1.23iT - 31T^{2} \)
37 \( 1 + 8.77iT - 37T^{2} \)
41 \( 1 - 0.979iT - 41T^{2} \)
43 \( 1 - 3.18T + 43T^{2} \)
47 \( 1 - 3.26T + 47T^{2} \)
53 \( 1 - 7.91T + 53T^{2} \)
59 \( 1 + 14.8iT - 59T^{2} \)
61 \( 1 + 9.53iT - 61T^{2} \)
67 \( 1 - 0.735T + 67T^{2} \)
71 \( 1 + 2.33T + 71T^{2} \)
73 \( 1 + 10.0T + 73T^{2} \)
79 \( 1 + 14.1iT - 79T^{2} \)
83 \( 1 - 4.23iT - 83T^{2} \)
89 \( 1 - 11.6iT - 89T^{2} \)
97 \( 1 - 4.84T + 97T^{2} \)
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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

−7.76113482706101432291297790499, −7.65381978819145310809741635764, −6.60957508761623345328592859566, −5.74263956667456223687774180434, −5.27591955074685336743104928530, −4.41625638034502947686545289667, −3.53372483860846214112646200662, −2.41086314828069537609587465134, −2.00702062701855149398194460148, −0.29648270441128732308810374175, 1.00553005088229000165310067622, 1.97554219773220097136611558736, 3.21395360525272090453997362904, 4.07719010360158992497821879127, 4.28530282446337209919974316321, 5.77278627527722959090188811006, 6.05142494351870878627083722568, 7.00470500867356632560892942746, 7.59825024122574061895511796435, 8.475485431177941173503253627950

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