Properties

Label 2-87e2-1.1-c1-0-19
Degree $2$
Conductor $7569$
Sign $1$
Analytic cond. $60.4387$
Root an. cond. $7.77423$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s + 3.82·4-s + 5-s − 2.82·7-s − 4.41·8-s − 2.41·10-s − 0.414·11-s − 3.82·13-s + 6.82·14-s + 2.99·16-s + 0.828·17-s − 6·19-s + 3.82·20-s + 0.999·22-s − 3.65·23-s − 4·25-s + 9.24·26-s − 10.8·28-s − 10.0·31-s + 1.58·32-s − 1.99·34-s − 2.82·35-s + 4·37-s + 14.4·38-s − 4.41·40-s − 4.48·41-s − 3.58·43-s + ⋯
L(s)  = 1  − 1.70·2-s + 1.91·4-s + 0.447·5-s − 1.06·7-s − 1.56·8-s − 0.763·10-s − 0.124·11-s − 1.06·13-s + 1.82·14-s + 0.749·16-s + 0.200·17-s − 1.37·19-s + 0.856·20-s + 0.213·22-s − 0.762·23-s − 0.800·25-s + 1.81·26-s − 2.04·28-s − 1.80·31-s + 0.280·32-s − 0.342·34-s − 0.478·35-s + 0.657·37-s + 2.34·38-s − 0.697·40-s − 0.700·41-s − 0.546·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7569\)    =    \(3^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(60.4387\)
Root analytic conductor: \(7.77423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7569,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1864434311\)
\(L(\frac12)\) \(\approx\) \(0.1864434311\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + 2.41T + 2T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 + 0.414T + 11T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 - 0.828T + 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 4.48T + 41T^{2} \)
43 \( 1 + 3.58T + 43T^{2} \)
47 \( 1 + 3.24T + 47T^{2} \)
53 \( 1 + 9.48T + 53T^{2} \)
59 \( 1 - 3.65T + 59T^{2} \)
61 \( 1 - 4.82T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 - 8.82T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 2.41T + 79T^{2} \)
83 \( 1 + 7.65T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + 4.48T + 97T^{2} \)
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

−8.027935450922638416745782373494, −7.30135217197974161253205441277, −6.69679164818859754824672607748, −6.15442611467651365906484249560, −5.34020596952404759033026834525, −4.20113052866169480744953467776, −3.20651914560291190074591658561, −2.26407692185047385096231872915, −1.76298780657070971386501607468, −0.26407704376819717144688877856, 0.26407704376819717144688877856, 1.76298780657070971386501607468, 2.26407692185047385096231872915, 3.20651914560291190074591658561, 4.20113052866169480744953467776, 5.34020596952404759033026834525, 6.15442611467651365906484249560, 6.69679164818859754824672607748, 7.30135217197974161253205441277, 8.027935450922638416745782373494

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