Properties

Label 2-5225-1.1-c1-0-119
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.324·2-s − 3.24·3-s − 1.89·4-s − 1.05·6-s − 0.166·7-s − 1.26·8-s + 7.51·9-s + 11-s + 6.14·12-s − 6.05·13-s − 0.0539·14-s + 3.37·16-s − 0.875·17-s + 2.44·18-s − 19-s + 0.538·21-s + 0.324·22-s − 4.61·23-s + 4.10·24-s − 1.96·26-s − 14.6·27-s + 0.314·28-s − 1.00·29-s + 8.35·31-s + 3.62·32-s − 3.24·33-s − 0.284·34-s + ⋯
L(s)  = 1  + 0.229·2-s − 1.87·3-s − 0.947·4-s − 0.430·6-s − 0.0627·7-s − 0.447·8-s + 2.50·9-s + 0.301·11-s + 1.77·12-s − 1.67·13-s − 0.0144·14-s + 0.844·16-s − 0.212·17-s + 0.575·18-s − 0.229·19-s + 0.117·21-s + 0.0692·22-s − 0.962·23-s + 0.837·24-s − 0.385·26-s − 2.82·27-s + 0.0594·28-s − 0.187·29-s + 1.50·31-s + 0.641·32-s − 0.564·33-s − 0.0487·34-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: \(1\)
Selberg data: \((2,\ 5225,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.324T + 2T^{2} \)
3 \( 1 + 3.24T + 3T^{2} \)
7 \( 1 + 0.166T + 7T^{2} \)
13 \( 1 + 6.05T + 13T^{2} \)
17 \( 1 + 0.875T + 17T^{2} \)
23 \( 1 + 4.61T + 23T^{2} \)
29 \( 1 + 1.00T + 29T^{2} \)
31 \( 1 - 8.35T + 31T^{2} \)
37 \( 1 - 7.03T + 37T^{2} \)
41 \( 1 + 8.32T + 41T^{2} \)
43 \( 1 + 6.62T + 43T^{2} \)
47 \( 1 - 8.43T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 4.98T + 59T^{2} \)
61 \( 1 - 7.32T + 61T^{2} \)
67 \( 1 - 0.360T + 67T^{2} \)
71 \( 1 + 7.74T + 71T^{2} \)
73 \( 1 - 6.39T + 73T^{2} \)
79 \( 1 + 0.309T + 79T^{2} \)
83 \( 1 - 3.15T + 83T^{2} \)
89 \( 1 - 13.0T + 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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

−7.67178255634889299818386096645, −6.91254065218095068082883688293, −6.23580438208749124828161248479, −5.60331210806864870909033075699, −4.82524299621819300934863476188, −4.56275940720477093378059302684, −3.67471101960691670550359778413, −2.23157621760551686982812017812, −0.869234884492166710893344445826, 0, 0.869234884492166710893344445826, 2.23157621760551686982812017812, 3.67471101960691670550359778413, 4.56275940720477093378059302684, 4.82524299621819300934863476188, 5.60331210806864870909033075699, 6.23580438208749124828161248479, 6.91254065218095068082883688293, 7.67178255634889299818386096645

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