Properties

Label 2-4536-1.1-c1-0-52
Degree $2$
Conductor $4536$
Sign $-1$
Analytic cond. $36.2201$
Root an. cond. $6.01831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.329·5-s − 7-s + 1.32·11-s + 3.07·13-s − 7.35·17-s − 2.93·19-s + 6.68·23-s − 4.89·25-s + 7.76·29-s − 3.27·31-s + 0.329·35-s + 0.329·37-s − 0.271·41-s − 10.9·43-s − 1.14·47-s + 49-s − 6.42·53-s − 0.437·55-s − 0.744·59-s + 8.84·61-s − 1.01·65-s + 8.57·67-s − 1.60·71-s − 13.4·73-s − 1.32·77-s − 1.25·79-s + 0.0632·83-s + ⋯
L(s)  = 1  − 0.147·5-s − 0.377·7-s + 0.400·11-s + 0.853·13-s − 1.78·17-s − 0.673·19-s + 1.39·23-s − 0.978·25-s + 1.44·29-s − 0.587·31-s + 0.0556·35-s + 0.0540·37-s − 0.0423·41-s − 1.67·43-s − 0.166·47-s + 0.142·49-s − 0.882·53-s − 0.0589·55-s − 0.0969·59-s + 1.13·61-s − 0.125·65-s + 1.04·67-s − 0.190·71-s − 1.57·73-s − 0.151·77-s − 0.141·79-s + 0.00694·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4536\)    =    \(2^{3} \cdot 3^{4} \cdot 7\)
Sign: $-1$
Analytic conductor: \(36.2201\)
Root analytic conductor: \(6.01831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4536,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 + 0.329T + 5T^{2} \)
11 \( 1 - 1.32T + 11T^{2} \)
13 \( 1 - 3.07T + 13T^{2} \)
17 \( 1 + 7.35T + 17T^{2} \)
19 \( 1 + 2.93T + 19T^{2} \)
23 \( 1 - 6.68T + 23T^{2} \)
29 \( 1 - 7.76T + 29T^{2} \)
31 \( 1 + 3.27T + 31T^{2} \)
37 \( 1 - 0.329T + 37T^{2} \)
41 \( 1 + 0.271T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 1.14T + 47T^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 + 0.744T + 59T^{2} \)
61 \( 1 - 8.84T + 61T^{2} \)
67 \( 1 - 8.57T + 67T^{2} \)
71 \( 1 + 1.60T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 + 1.25T + 79T^{2} \)
83 \( 1 - 0.0632T + 83T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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

−8.197585682759533761113833767859, −6.93880485717161910567929122333, −6.68519164319607493319841199971, −5.91164009741976607595558723382, −4.88187321047248879527561899882, −4.19787190768855756528712785831, −3.40259187707984308290127433524, −2.45064058623443354106003522833, −1.39102281670618550980614523197, 0, 1.39102281670618550980614523197, 2.45064058623443354106003522833, 3.40259187707984308290127433524, 4.19787190768855756528712785831, 4.88187321047248879527561899882, 5.91164009741976607595558723382, 6.68519164319607493319841199971, 6.93880485717161910567929122333, 8.197585682759533761113833767859

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