Properties

Label 2-5550-1.1-c1-0-41
Degree $2$
Conductor $5550$
Sign $1$
Analytic cond. $44.3169$
Root an. cond. $6.65709$
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-s − 3-s + 4-s + 6-s + 3.46·7-s − 8-s + 9-s + 4.15·11-s − 12-s + 3.73·13-s − 3.46·14-s + 16-s − 4.63·17-s − 18-s + 5.88·19-s − 3.46·21-s − 4.15·22-s − 4.36·23-s + 24-s − 3.73·26-s − 27-s + 3.46·28-s + 8.03·29-s + 0.895·31-s − 32-s − 4.15·33-s + 4.63·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.30·7-s − 0.353·8-s + 0.333·9-s + 1.25·11-s − 0.288·12-s + 1.03·13-s − 0.926·14-s + 0.250·16-s − 1.12·17-s − 0.235·18-s + 1.34·19-s − 0.756·21-s − 0.885·22-s − 0.909·23-s + 0.204·24-s − 0.733·26-s − 0.192·27-s + 0.654·28-s + 1.49·29-s + 0.160·31-s − 0.176·32-s − 0.723·33-s + 0.794·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5550\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 37\)
Sign: $1$
Analytic conductor: \(44.3169\)
Root analytic conductor: \(6.65709\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.733499582\)
\(L(\frac12)\) \(\approx\) \(1.733499582\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
37 \( 1 - T \)
good7 \( 1 - 3.46T + 7T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
13 \( 1 - 3.73T + 13T^{2} \)
17 \( 1 + 4.63T + 17T^{2} \)
19 \( 1 - 5.88T + 19T^{2} \)
23 \( 1 + 4.36T + 23T^{2} \)
29 \( 1 - 8.03T + 29T^{2} \)
31 \( 1 - 0.895T + 31T^{2} \)
41 \( 1 + 2.89T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 - 6.78T + 47T^{2} \)
53 \( 1 - 8.63T + 53T^{2} \)
59 \( 1 - 6.93T + 59T^{2} \)
61 \( 1 + 2.20T + 61T^{2} \)
67 \( 1 - 6.09T + 67T^{2} \)
71 \( 1 + 13.1T + 71T^{2} \)
73 \( 1 - 7.19T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 7.06T + 97T^{2} \)
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

−8.344879216440505112341823815918, −7.42831316722326391535076090152, −6.80320970906022861265370652393, −6.08710113366665458314298997586, −5.41180118180413728835689362746, −4.41283997663094533202593594826, −3.88961624172070278318701429624, −2.54405409308194698593413306208, −1.48417783057371004061027479295, −0.925558155661998755991364751695, 0.925558155661998755991364751695, 1.48417783057371004061027479295, 2.54405409308194698593413306208, 3.88961624172070278318701429624, 4.41283997663094533202593594826, 5.41180118180413728835689362746, 6.08710113366665458314298997586, 6.80320970906022861265370652393, 7.42831316722326391535076090152, 8.344879216440505112341823815918

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