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Properties

Label 2-72e2-8.5-c1-0-54
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  − 1.73i·5-s + 3.61·7-s − 0.734i·11-s − 0.609i·13-s + 5.52·17-s + 2i·19-s + 4.73·23-s + 2.00·25-s + 7.83i·29-s − 9.41·31-s − 6.25i·35-s + 2.34i·37-s + 8.52·41-s + 10.2i·43-s + 11.7·47-s + ⋯
L(s)  = 1  − 0.774i·5-s + 1.36·7-s − 0.221i·11-s − 0.169i·13-s + 1.33·17-s + 0.458i·19-s + 0.987·23-s + 0.400·25-s + 1.45i·29-s − 1.69·31-s − 1.05i·35-s + 0.384i·37-s + 1.33·41-s + 1.56i·43-s + 1.71·47-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: $\chi_{5184} (2593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5184,\ (\ :1/2),\ 0.965 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.632663059\)
\(L(\frac12)\) \(\approx\) \(2.632663059\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.61T + 7T^{2} \)
11 \( 1 + 0.734iT - 11T^{2} \)
13 \( 1 + 0.609iT - 13T^{2} \)
17 \( 1 - 5.52T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 - 4.73T + 23T^{2} \)
29 \( 1 - 7.83iT - 29T^{2} \)
31 \( 1 + 9.41T + 31T^{2} \)
37 \( 1 - 2.34iT - 37T^{2} \)
41 \( 1 - 8.52T + 41T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + 13.0iT - 53T^{2} \)
59 \( 1 - 1.20iT - 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 + 6.31iT - 67T^{2} \)
71 \( 1 + 2.63T + 71T^{2} \)
73 \( 1 + 2.05T + 73T^{2} \)
79 \( 1 - 2.49T + 79T^{2} \)
83 \( 1 - 12.2iT - 83T^{2} \)
89 \( 1 - 1.94T + 89T^{2} \)
97 \( 1 + 15.5T + 97T^{2} \)
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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

−8.136937706011489603896750431262, −7.63655374575917053762182000808, −6.91173785816293684234491364539, −5.65228525521903227030158106349, −5.32713029324441317400327208148, −4.64276489632837592105105033285, −3.75908749073140880305992424836, −2.81679148557306438096113187637, −1.53724134964069856032267174280, −1.02437140318962088342965296932, 0.909925093846414396776528693836, 1.99950505746416678884163359544, 2.78856211513326535314365198184, 3.79172045498339632596712959612, 4.54970648829569612454877848409, 5.41174445631963234512796427054, 5.92621351592753190205941737443, 7.16197146145094541306381541861, 7.36710573610739514473693460488, 8.098757154731334963379010785861

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