Properties

Label 2-72e2-8.5-c1-0-18
Degree $2$
Conductor $5184$
Sign $0.707 - 0.707i$
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  − 1.73i·5-s − 3.46·7-s − 1.73i·13-s + 3·17-s + 2i·19-s − 6.92·23-s + 2.00·25-s + 8.66i·29-s − 3.46·31-s + 5.99i·35-s − 8.66i·37-s − 6·41-s + 4i·43-s + 3.46·47-s + 4.99·49-s + ⋯
L(s)  = 1  − 0.774i·5-s − 1.30·7-s − 0.480i·13-s + 0.727·17-s + 0.458i·19-s − 1.44·23-s + 0.400·25-s + 1.60i·29-s − 0.622·31-s + 1.01i·35-s − 1.42i·37-s − 0.937·41-s + 0.609i·43-s + 0.505·47-s + 0.714·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(41.3944\)
Root analytic conductor: \(6.43385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5184} (2593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.026892298\)
\(L(\frac12)\) \(\approx\) \(1.026892298\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.73iT - 5T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 6.92T + 23T^{2} \)
29 \( 1 - 8.66iT - 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + 8.66iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 6.92iT - 53T^{2} \)
59 \( 1 + 12iT - 59T^{2} \)
61 \( 1 - 5.19iT - 61T^{2} \)
67 \( 1 - 10iT - 67T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 + 6.92T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−8.368473556461435432210937627210, −7.57588838751890288501364558037, −6.89904407818229285659357320835, −5.97676844471004406811245754097, −5.56172874512866653600924338355, −4.64647681968424460075705132195, −3.65590546536196443837190068297, −3.17616711571042104923054492612, −1.96029550852957378924245978869, −0.811170074160528653573777228183, 0.35504717286255685208212360787, 1.92839726827070503714512190686, 2.85527042338479343629720592059, 3.49878094511729205532547051818, 4.24677521253612454565948013894, 5.33217258365069066346072014921, 6.24233125827684609505635427679, 6.53566734918750165000942861191, 7.31724847569248112038824747298, 8.030490682930519164611479411124

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