Properties

Label 2-5225-1.1-c1-0-13
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  − 1.98·2-s − 0.104·3-s + 1.95·4-s + 0.208·6-s + 1.24·7-s + 0.0977·8-s − 2.98·9-s − 11-s − 0.204·12-s − 2.45·13-s − 2.47·14-s − 4.09·16-s − 8.07·17-s + 5.94·18-s + 19-s − 0.130·21-s + 1.98·22-s − 0.595·23-s − 0.0102·24-s + 4.87·26-s + 0.628·27-s + 2.42·28-s − 6.20·29-s − 9.51·31-s + 7.94·32-s + 0.104·33-s + 16.0·34-s + ⋯
L(s)  = 1  − 1.40·2-s − 0.0605·3-s + 0.975·4-s + 0.0851·6-s + 0.470·7-s + 0.0345·8-s − 0.996·9-s − 0.301·11-s − 0.0590·12-s − 0.680·13-s − 0.661·14-s − 1.02·16-s − 1.95·17-s + 1.40·18-s + 0.229·19-s − 0.0285·21-s + 0.423·22-s − 0.124·23-s − 0.00209·24-s + 0.956·26-s + 0.120·27-s + 0.459·28-s − 1.15·29-s − 1.70·31-s + 1.40·32-s + 0.0182·33-s + 2.75·34-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

\(L(1)\) \(\approx\) \(0.3063978830\)
\(L(\frac12)\) \(\approx\) \(0.3063978830\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.98T + 2T^{2} \)
3 \( 1 + 0.104T + 3T^{2} \)
7 \( 1 - 1.24T + 7T^{2} \)
13 \( 1 + 2.45T + 13T^{2} \)
17 \( 1 + 8.07T + 17T^{2} \)
23 \( 1 + 0.595T + 23T^{2} \)
29 \( 1 + 6.20T + 29T^{2} \)
31 \( 1 + 9.51T + 31T^{2} \)
37 \( 1 - 6.49T + 37T^{2} \)
41 \( 1 + 8.06T + 41T^{2} \)
43 \( 1 + 2.33T + 43T^{2} \)
47 \( 1 + 8.72T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 - 0.601T + 59T^{2} \)
61 \( 1 + 1.23T + 61T^{2} \)
67 \( 1 + 5.79T + 67T^{2} \)
71 \( 1 - 7.74T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 - 1.79T + 79T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 + 7.25T + 89T^{2} \)
97 \( 1 - 8.72T + 97T^{2} \)
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

−8.302293533314118627626825267148, −7.70253966348309305624727876292, −7.02319290407593275396036066629, −6.30924342498001836431520708439, −5.26731972017301912903948310470, −4.69658696918885168064481817089, −3.57700904478818591507529357273, −2.33761234012995124377234892765, −1.87069116890168818139451454329, −0.35358057330154112375507261784, 0.35358057330154112375507261784, 1.87069116890168818139451454329, 2.33761234012995124377234892765, 3.57700904478818591507529357273, 4.69658696918885168064481817089, 5.26731972017301912903948310470, 6.30924342498001836431520708439, 7.02319290407593275396036066629, 7.70253966348309305624727876292, 8.302293533314118627626825267148

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