Properties

Label 2-688-1.1-c5-0-83
Degree $2$
Conductor $688$
Sign $-1$
Analytic cond. $110.344$
Root an. cond. $10.5044$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.05·3-s + 27.7·5-s + 103.·7-s − 233.·9-s + 158.·11-s − 578.·13-s − 84.7·15-s + 253.·17-s + 3.09e3·19-s − 317.·21-s − 4.16e3·23-s − 2.35e3·25-s + 1.45e3·27-s + 6.77e3·29-s − 6.26e3·31-s − 484.·33-s + 2.87e3·35-s − 3.29e3·37-s + 1.77e3·39-s − 6.15e3·41-s + 1.84e3·43-s − 6.47e3·45-s − 8.15e3·47-s − 6.05e3·49-s − 776.·51-s − 3.04e4·53-s + 4.38e3·55-s + ⋯
L(s)  = 1  − 0.196·3-s + 0.495·5-s + 0.799·7-s − 0.961·9-s + 0.394·11-s − 0.950·13-s − 0.0972·15-s + 0.213·17-s + 1.96·19-s − 0.156·21-s − 1.64·23-s − 0.754·25-s + 0.384·27-s + 1.49·29-s − 1.17·31-s − 0.0774·33-s + 0.396·35-s − 0.395·37-s + 0.186·39-s − 0.571·41-s + 0.152·43-s − 0.476·45-s − 0.538·47-s − 0.360·49-s − 0.0418·51-s − 1.48·53-s + 0.195·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(688\)    =    \(2^{4} \cdot 43\)
Sign: $-1$
Analytic conductor: \(110.344\)
Root analytic conductor: \(10.5044\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{688} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 688,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 - 1.84e3T \)
good3 \( 1 + 3.05T + 243T^{2} \)
5 \( 1 - 27.7T + 3.12e3T^{2} \)
7 \( 1 - 103.T + 1.68e4T^{2} \)
11 \( 1 - 158.T + 1.61e5T^{2} \)
13 \( 1 + 578.T + 3.71e5T^{2} \)
17 \( 1 - 253.T + 1.41e6T^{2} \)
19 \( 1 - 3.09e3T + 2.47e6T^{2} \)
23 \( 1 + 4.16e3T + 6.43e6T^{2} \)
29 \( 1 - 6.77e3T + 2.05e7T^{2} \)
31 \( 1 + 6.26e3T + 2.86e7T^{2} \)
37 \( 1 + 3.29e3T + 6.93e7T^{2} \)
41 \( 1 + 6.15e3T + 1.15e8T^{2} \)
47 \( 1 + 8.15e3T + 2.29e8T^{2} \)
53 \( 1 + 3.04e4T + 4.18e8T^{2} \)
59 \( 1 - 4.52e4T + 7.14e8T^{2} \)
61 \( 1 + 7.25e3T + 8.44e8T^{2} \)
67 \( 1 + 1.96e4T + 1.35e9T^{2} \)
71 \( 1 - 4.81e4T + 1.80e9T^{2} \)
73 \( 1 - 4.25e4T + 2.07e9T^{2} \)
79 \( 1 - 6.51e4T + 3.07e9T^{2} \)
83 \( 1 + 7.04e4T + 3.93e9T^{2} \)
89 \( 1 - 2.96e4T + 5.58e9T^{2} \)
97 \( 1 + 8.94e4T + 8.58e9T^{2} \)
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

−9.463025411276367335278228368422, −8.290215524887779101092537532362, −7.67151285577936716924184312392, −6.52192296928151237804506757012, −5.51765693211709914990207965794, −4.97925334737494941785535255722, −3.60114764404426453520576059772, −2.43054357564583299476877726383, −1.37474867488948856245032918495, 0, 1.37474867488948856245032918495, 2.43054357564583299476877726383, 3.60114764404426453520576059772, 4.97925334737494941785535255722, 5.51765693211709914990207965794, 6.52192296928151237804506757012, 7.67151285577936716924184312392, 8.290215524887779101092537532362, 9.463025411276367335278228368422

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