Properties

Label 2-5225-1.1-c1-0-142
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  + 0.527·2-s − 1.24·3-s − 1.72·4-s − 0.656·6-s − 3.22·7-s − 1.96·8-s − 1.45·9-s + 11-s + 2.14·12-s + 4.87·13-s − 1.69·14-s + 2.41·16-s − 7.11·17-s − 0.764·18-s + 19-s + 4.01·21-s + 0.527·22-s − 1.12·23-s + 2.44·24-s + 2.56·26-s + 5.53·27-s + 5.54·28-s + 8.47·29-s − 0.926·31-s + 5.19·32-s − 1.24·33-s − 3.75·34-s + ⋯
L(s)  = 1  + 0.372·2-s − 0.718·3-s − 0.861·4-s − 0.267·6-s − 1.21·7-s − 0.693·8-s − 0.483·9-s + 0.301·11-s + 0.618·12-s + 1.35·13-s − 0.453·14-s + 0.602·16-s − 1.72·17-s − 0.180·18-s + 0.229·19-s + 0.875·21-s + 0.112·22-s − 0.234·23-s + 0.498·24-s + 0.503·26-s + 1.06·27-s + 1.04·28-s + 1.57·29-s − 0.166·31-s + 0.918·32-s − 0.216·33-s − 0.643·34-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 \( 1 - 0.527T + 2T^{2} \)
3 \( 1 + 1.24T + 3T^{2} \)
7 \( 1 + 3.22T + 7T^{2} \)
13 \( 1 - 4.87T + 13T^{2} \)
17 \( 1 + 7.11T + 17T^{2} \)
23 \( 1 + 1.12T + 23T^{2} \)
29 \( 1 - 8.47T + 29T^{2} \)
31 \( 1 + 0.926T + 31T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 - 7.49T + 41T^{2} \)
43 \( 1 - 7.22T + 43T^{2} \)
47 \( 1 + 12.4T + 47T^{2} \)
53 \( 1 + 7.92T + 53T^{2} \)
59 \( 1 - 1.91T + 59T^{2} \)
61 \( 1 + 11.5T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 0.979T + 71T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 + 3.05T + 79T^{2} \)
83 \( 1 - 2.60T + 83T^{2} \)
89 \( 1 - 5.38T + 89T^{2} \)
97 \( 1 + 15.6T + 97T^{2} \)
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

−8.008673166298826607309048462151, −6.69387032436688384336911782187, −6.16174379399124792406094367991, −5.96899685855999758001447152420, −4.78176790486215781689974225560, −4.28014787743017484488657647331, −3.38302069549451511399520385991, −2.67707971515479932748775502007, −0.981885962437332969368780517883, 0, 0.981885962437332969368780517883, 2.67707971515479932748775502007, 3.38302069549451511399520385991, 4.28014787743017484488657647331, 4.78176790486215781689974225560, 5.96899685855999758001447152420, 6.16174379399124792406094367991, 6.69387032436688384336911782187, 8.008673166298826607309048462151

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