Properties

Label 2-5225-1.1-c1-0-9
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.719·2-s − 0.163·3-s − 1.48·4-s + 0.117·6-s − 1.05·7-s + 2.50·8-s − 2.97·9-s + 11-s + 0.241·12-s − 3.88·13-s + 0.759·14-s + 1.16·16-s − 5.65·17-s + 2.13·18-s − 19-s + 0.172·21-s − 0.719·22-s − 2.78·23-s − 0.408·24-s + 2.79·26-s + 0.974·27-s + 1.56·28-s + 0.890·29-s + 6.13·31-s − 5.84·32-s − 0.163·33-s + 4.06·34-s + ⋯
L(s)  = 1  − 0.508·2-s − 0.0941·3-s − 0.741·4-s + 0.0478·6-s − 0.399·7-s + 0.885·8-s − 0.991·9-s + 0.301·11-s + 0.0698·12-s − 1.07·13-s + 0.203·14-s + 0.291·16-s − 1.37·17-s + 0.503·18-s − 0.229·19-s + 0.0376·21-s − 0.153·22-s − 0.581·23-s − 0.0833·24-s + 0.548·26-s + 0.187·27-s + 0.296·28-s + 0.165·29-s + 1.10·31-s − 1.03·32-s − 0.0283·33-s + 0.696·34-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

\(L(1)\) \(\approx\) \(0.3325520632\)
\(L(\frac12)\) \(\approx\) \(0.3325520632\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.719T + 2T^{2} \)
3 \( 1 + 0.163T + 3T^{2} \)
7 \( 1 + 1.05T + 7T^{2} \)
13 \( 1 + 3.88T + 13T^{2} \)
17 \( 1 + 5.65T + 17T^{2} \)
23 \( 1 + 2.78T + 23T^{2} \)
29 \( 1 - 0.890T + 29T^{2} \)
31 \( 1 - 6.13T + 31T^{2} \)
37 \( 1 + 3.51T + 37T^{2} \)
41 \( 1 + 2.28T + 41T^{2} \)
43 \( 1 + 3.91T + 43T^{2} \)
47 \( 1 + 7.75T + 47T^{2} \)
53 \( 1 + 7.61T + 53T^{2} \)
59 \( 1 + 2.17T + 59T^{2} \)
61 \( 1 - 0.475T + 61T^{2} \)
67 \( 1 - 0.383T + 67T^{2} \)
71 \( 1 + 2.51T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + 4.27T + 79T^{2} \)
83 \( 1 - 1.39T + 83T^{2} \)
89 \( 1 - 8.36T + 89T^{2} \)
97 \( 1 - 6.75T + 97T^{2} \)
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

−8.324764308312224283180193842775, −7.71259287434811994685165513557, −6.69223278501219247262162411581, −6.22988651403582027512082680249, −5.07699849310555826702043511829, −4.70412133241742141194471412441, −3.72697559552624861937951789182, −2.79382926303924304795272338099, −1.80765296153194340259737760598, −0.32843211756708005098351951043, 0.32843211756708005098351951043, 1.80765296153194340259737760598, 2.79382926303924304795272338099, 3.72697559552624861937951789182, 4.70412133241742141194471412441, 5.07699849310555826702043511829, 6.22988651403582027512082680249, 6.69223278501219247262162411581, 7.71259287434811994685165513557, 8.324764308312224283180193842775

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