Properties

Label 2-72e2-8.5-c1-0-38
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·5-s − 0.990·7-s − 2.10i·11-s + 6.36i·13-s − 3.81·17-s + 2i·19-s + 7.10·23-s + 2.00·25-s − 8.34i·29-s + 2.15·31-s + 1.71i·35-s − 4.62i·37-s − 0.816·41-s + 2.28i·43-s − 6.78·47-s + ⋯
L(s)  = 1  − 0.774i·5-s − 0.374·7-s − 0.633i·11-s + 1.76i·13-s − 0.925·17-s + 0.458i·19-s + 1.48·23-s + 0.400·25-s − 1.54i·29-s + 0.387·31-s + 0.290i·35-s − 0.761i·37-s − 0.127·41-s + 0.348i·43-s − 0.989·47-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

\(L(1)\) \(\approx\) \(1.702401273\)
\(L(\frac12)\) \(\approx\) \(1.702401273\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.990T + 7T^{2} \)
11 \( 1 + 2.10iT - 11T^{2} \)
13 \( 1 - 6.36iT - 13T^{2} \)
17 \( 1 + 3.81T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 - 7.10T + 23T^{2} \)
29 \( 1 + 8.34iT - 29T^{2} \)
31 \( 1 - 2.15T + 31T^{2} \)
37 \( 1 + 4.62iT - 37T^{2} \)
41 \( 1 + 0.816T + 41T^{2} \)
43 \( 1 - 2.28iT - 43T^{2} \)
47 \( 1 + 6.78T + 47T^{2} \)
53 \( 1 - 3.14iT - 53T^{2} \)
59 \( 1 - 11.9iT - 59T^{2} \)
61 \( 1 + 4.87iT - 61T^{2} \)
67 \( 1 - 13.7iT - 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + 9.08T + 79T^{2} \)
83 \( 1 - 4.28iT - 83T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.378407664675898959075923703011, −7.43270901629134667811907716958, −6.61320784489182439957747981209, −6.19332591009156632973528063357, −5.14400160392706261666714055330, −4.49378563397833695354677594378, −3.83970041804895743323554289380, −2.75929612846528770944374301561, −1.79892129219009776775352599165, −0.71215079956613412659487756453, 0.69705600768052912458525341211, 2.07323284975587755308245449176, 3.18631711061104938001727992848, 3.27349284463407361905691825294, 4.88482429317821487384835224047, 5.06441232377944857124354217252, 6.35579324868028367993386872375, 6.75004894265260720313145341041, 7.41287215817303717721398595776, 8.212213221579521338816058975440

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