Properties

Label 2-39e2-1.1-c3-0-75
Degree $2$
Conductor $1521$
Sign $1$
Analytic cond. $89.7419$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.69·2-s + 5.63·4-s + 18.7·5-s − 24.1·7-s + 8.74·8-s − 69.2·10-s + 49.2·11-s + 89.1·14-s − 77.3·16-s + 65.3·17-s + 109.·19-s + 105.·20-s − 181.·22-s + 83.2·23-s + 226.·25-s − 135.·28-s − 4.99·29-s − 255.·31-s + 215.·32-s − 241.·34-s − 452.·35-s + 93.6·37-s − 405.·38-s + 164.·40-s − 67.9·41-s + 142.·43-s + 277.·44-s + ⋯
L(s)  = 1  − 1.30·2-s + 0.703·4-s + 1.67·5-s − 1.30·7-s + 0.386·8-s − 2.19·10-s + 1.35·11-s + 1.70·14-s − 1.20·16-s + 0.931·17-s + 1.32·19-s + 1.18·20-s − 1.76·22-s + 0.755·23-s + 1.81·25-s − 0.917·28-s − 0.0319·29-s − 1.48·31-s + 1.19·32-s − 1.21·34-s − 2.18·35-s + 0.416·37-s − 1.73·38-s + 0.648·40-s − 0.258·41-s + 0.505·43-s + 0.950·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(89.7419\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.586495337\)
\(L(\frac12)\) \(\approx\) \(1.586495337\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + 3.69T + 8T^{2} \)
5 \( 1 - 18.7T + 125T^{2} \)
7 \( 1 + 24.1T + 343T^{2} \)
11 \( 1 - 49.2T + 1.33e3T^{2} \)
17 \( 1 - 65.3T + 4.91e3T^{2} \)
19 \( 1 - 109.T + 6.85e3T^{2} \)
23 \( 1 - 83.2T + 1.21e4T^{2} \)
29 \( 1 + 4.99T + 2.43e4T^{2} \)
31 \( 1 + 255.T + 2.97e4T^{2} \)
37 \( 1 - 93.6T + 5.06e4T^{2} \)
41 \( 1 + 67.9T + 6.89e4T^{2} \)
43 \( 1 - 142.T + 7.95e4T^{2} \)
47 \( 1 - 379.T + 1.03e5T^{2} \)
53 \( 1 - 389.T + 1.48e5T^{2} \)
59 \( 1 + 133.T + 2.05e5T^{2} \)
61 \( 1 - 620.T + 2.26e5T^{2} \)
67 \( 1 - 119.T + 3.00e5T^{2} \)
71 \( 1 + 361.T + 3.57e5T^{2} \)
73 \( 1 + 748.T + 3.89e5T^{2} \)
79 \( 1 - 514.T + 4.93e5T^{2} \)
83 \( 1 + 260.T + 5.71e5T^{2} \)
89 \( 1 - 833.T + 7.04e5T^{2} \)
97 \( 1 + 1.48e3T + 9.12e5T^{2} \)
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

−9.263734442196759233672546722242, −8.804532142667475220411158078780, −7.39842878737965249517602243846, −6.85509022208007821436459804523, −6.00742706809656961135144746886, −5.30300008061938726705611964340, −3.77819127038899332581154674774, −2.70321509999014958749612960624, −1.53098363887141625123271184495, −0.826147854525943561136768423378, 0.826147854525943561136768423378, 1.53098363887141625123271184495, 2.70321509999014958749612960624, 3.77819127038899332581154674774, 5.30300008061938726705611964340, 6.00742706809656961135144746886, 6.85509022208007821436459804523, 7.39842878737965249517602243846, 8.804532142667475220411158078780, 9.263734442196759233672546722242

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