Properties

Label 2-72e2-1.1-c1-0-14
Degree $2$
Conductor $5184$
Sign $1$
Analytic cond. $41.3944$
Root an. cond. $6.43385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.46·5-s + 6.46·13-s − 5.92·17-s + 1.07·25-s + 1.53·29-s + 9.39·37-s − 10·41-s − 7·49-s + 14·53-s − 15.3·61-s − 15.9·65-s + 16.8·73-s + 14.6·85-s + 18.8·89-s + 18·97-s − 2·101-s − 14.3·109-s − 20.8·113-s + ⋯
L(s)  = 1  − 1.10·5-s + 1.79·13-s − 1.43·17-s + 0.214·25-s + 0.285·29-s + 1.54·37-s − 1.56·41-s − 49-s + 1.92·53-s − 1.97·61-s − 1.97·65-s + 1.97·73-s + 1.58·85-s + 1.99·89-s + 1.82·97-s − 0.199·101-s − 1.37·109-s − 1.96·113-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.380217114\)
\(L(\frac12)\) \(\approx\) \(1.380217114\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.46T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.46T + 13T^{2} \)
17 \( 1 + 5.92T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 1.53T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 9.39T + 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 14T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 16.8T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 18.8T + 89T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−8.183373763893933878074467430087, −7.66885476172907097388010508783, −6.61640249578787500179034526634, −6.31618041199326848817890836310, −5.22529048439444723504808027884, −4.30660659411756592486264260830, −3.83992892937984209445472119272, −3.01613638117295475511857923017, −1.83449713141081606275191208314, −0.63949082670133113016235101328, 0.63949082670133113016235101328, 1.83449713141081606275191208314, 3.01613638117295475511857923017, 3.83992892937984209445472119272, 4.30660659411756592486264260830, 5.22529048439444723504808027884, 6.31618041199326848817890836310, 6.61640249578787500179034526634, 7.66885476172907097388010508783, 8.183373763893933878074467430087

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