Properties

Label 2-5225-1.1-c1-0-121
Degree $2$
Conductor $5225$
Sign $1$
Analytic cond. $41.7218$
Root an. cond. $6.45924$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.66·2-s + 1.25·3-s + 0.769·4-s − 2.09·6-s + 4.13·7-s + 2.04·8-s − 1.41·9-s + 11-s + 0.969·12-s + 3.80·13-s − 6.88·14-s − 4.94·16-s − 2.64·17-s + 2.35·18-s + 19-s + 5.20·21-s − 1.66·22-s + 5.58·23-s + 2.57·24-s − 6.33·26-s − 5.55·27-s + 3.18·28-s − 6.34·29-s + 7.53·31-s + 4.13·32-s + 1.25·33-s + 4.39·34-s + ⋯
L(s)  = 1  − 1.17·2-s + 0.726·3-s + 0.384·4-s − 0.855·6-s + 1.56·7-s + 0.723·8-s − 0.471·9-s + 0.301·11-s + 0.279·12-s + 1.05·13-s − 1.83·14-s − 1.23·16-s − 0.640·17-s + 0.555·18-s + 0.229·19-s + 1.13·21-s − 0.354·22-s + 1.16·23-s + 0.525·24-s − 1.24·26-s − 1.06·27-s + 0.601·28-s − 1.17·29-s + 1.35·31-s + 0.731·32-s + 0.219·33-s + 0.754·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5225\)    =    \(5^{2} \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(41.7218\)
Root analytic conductor: \(6.45924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.680692090\)
\(L(\frac12)\) \(\approx\) \(1.680692090\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 + 1.66T + 2T^{2} \)
3 \( 1 - 1.25T + 3T^{2} \)
7 \( 1 - 4.13T + 7T^{2} \)
13 \( 1 - 3.80T + 13T^{2} \)
17 \( 1 + 2.64T + 17T^{2} \)
23 \( 1 - 5.58T + 23T^{2} \)
29 \( 1 + 6.34T + 29T^{2} \)
31 \( 1 - 7.53T + 31T^{2} \)
37 \( 1 - 3.00T + 37T^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + 6.54T + 47T^{2} \)
53 \( 1 - 7.14T + 53T^{2} \)
59 \( 1 + 7.34T + 59T^{2} \)
61 \( 1 - 4.72T + 61T^{2} \)
67 \( 1 - 7.94T + 67T^{2} \)
71 \( 1 - 9.69T + 71T^{2} \)
73 \( 1 + 16.9T + 73T^{2} \)
79 \( 1 - 3.09T + 79T^{2} \)
83 \( 1 + 1.51T + 83T^{2} \)
89 \( 1 - 8.71T + 89T^{2} \)
97 \( 1 - 6.68T + 97T^{2} \)
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

−8.301193887977065025071357907046, −7.84113850641425422985983892311, −7.18734800208540035344673357397, −6.20365273459857074884090592319, −5.21630005487027909216909181086, −4.49453941692384015368445711721, −3.66804929827233011014680392521, −2.50625362706656426231885286789, −1.68859702361214157007263340083, −0.873705550631303583248271189942, 0.873705550631303583248271189942, 1.68859702361214157007263340083, 2.50625362706656426231885286789, 3.66804929827233011014680392521, 4.49453941692384015368445711721, 5.21630005487027909216909181086, 6.20365273459857074884090592319, 7.18734800208540035344673357397, 7.84113850641425422985983892311, 8.301193887977065025071357907046

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