Properties

Label 2-72e2-24.11-c1-0-59
Degree $2$
Conductor $5184$
Sign $0.965 + 0.258i$
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  + 1.59·5-s − 1.16i·7-s + 1.98i·11-s + 0.164i·13-s − 3.57i·17-s + 3.16·19-s + 1.21·23-s − 2.44·25-s + 4.98·29-s + 6.64i·31-s − 1.85i·35-s − 1.10i·37-s − 2.02i·41-s + 8.34·43-s − 2.21·47-s + ⋯
L(s)  = 1  + 0.714·5-s − 0.440i·7-s + 0.597i·11-s + 0.0455i·13-s − 0.867i·17-s + 0.725·19-s + 0.253·23-s − 0.489·25-s + 0.925·29-s + 1.19i·31-s − 0.314i·35-s − 0.181i·37-s − 0.316i·41-s + 1.27·43-s − 0.323·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(5184\)    =    \(2^{6} \cdot 3^{4}\)
Sign: $0.965 + 0.258i$
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: \((2,\ 5184,\ (\ :1/2),\ 0.965 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.331852755\)
\(L(\frac12)\) \(\approx\) \(2.331852755\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.59T + 5T^{2} \)
7 \( 1 + 1.16iT - 7T^{2} \)
11 \( 1 - 1.98iT - 11T^{2} \)
13 \( 1 - 0.164iT - 13T^{2} \)
17 \( 1 + 3.57iT - 17T^{2} \)
19 \( 1 - 3.16T + 19T^{2} \)
23 \( 1 - 1.21T + 23T^{2} \)
29 \( 1 - 4.98T + 29T^{2} \)
31 \( 1 - 6.64iT - 31T^{2} \)
37 \( 1 + 1.10iT - 37T^{2} \)
41 \( 1 + 2.02iT - 41T^{2} \)
43 \( 1 - 8.34T + 43T^{2} \)
47 \( 1 + 2.21T + 47T^{2} \)
53 \( 1 + 9.65T + 53T^{2} \)
59 \( 1 + 0.888iT - 59T^{2} \)
61 \( 1 + 9.74iT - 61T^{2} \)
67 \( 1 - 3.98T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 - 0.879T + 73T^{2} \)
79 \( 1 - 2.91iT - 79T^{2} \)
83 \( 1 - 13.9iT - 83T^{2} \)
89 \( 1 + 8.41iT - 89T^{2} \)
97 \( 1 - 4.31T + 97T^{2} \)
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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

−8.130575485642006195231907886743, −7.34710856949679658925211332329, −6.83881337884632543934075966770, −6.03607346403273660204416087668, −5.18902417944002685955752438909, −4.65798079069937062988985451195, −3.63686432990809203143293497138, −2.75488713780447143108221926333, −1.85291591291755721933988230716, −0.800987921106885069141632984949, 0.897695623420779653313694258016, 1.99156023458084062213281151016, 2.79672347019827950222043487765, 3.69684613502032844671309350027, 4.61259566495934234782872495592, 5.55646674048077171474104682313, 5.97847190619511519920930666695, 6.62867004609732029989131736253, 7.67449513966416650517454108347, 8.193581404023602067077185184941

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