Properties

Label 2-95e2-1.1-c1-0-212
Degree $2$
Conductor $9025$
Sign $-1$
Analytic cond. $72.0649$
Root an. cond. $8.48910$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 2·7-s + 3·8-s − 3·9-s − 4·11-s − 2·13-s + 2·14-s − 16-s − 4·17-s + 3·18-s + 4·22-s + 6·23-s + 2·26-s + 2·28-s + 6·29-s + 4·31-s − 5·32-s + 4·34-s + 3·36-s − 10·37-s + 10·41-s − 2·43-s + 4·44-s − 6·46-s + 6·47-s − 3·49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s − 9-s − 1.20·11-s − 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.970·17-s + 0.707·18-s + 0.852·22-s + 1.25·23-s + 0.392·26-s + 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.883·32-s + 0.685·34-s + 1/2·36-s − 1.64·37-s + 1.56·41-s − 0.304·43-s + 0.603·44-s − 0.884·46-s + 0.875·47-s − 3/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(72.0649\)
Root analytic conductor: \(8.48910\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9025,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−7.50136799669702123180560377799, −6.87231635746065961134156986184, −6.07990432071003204598796847667, −5.15049393566020071335799991802, −4.84132098018455846662836521172, −3.78973024595814975369272522800, −2.84788762764959730109880934230, −2.32358969255822247087366120152, −0.816254076128081240605604169498, 0, 0.816254076128081240605604169498, 2.32358969255822247087366120152, 2.84788762764959730109880934230, 3.78973024595814975369272522800, 4.84132098018455846662836521172, 5.15049393566020071335799991802, 6.07990432071003204598796847667, 6.87231635746065961134156986184, 7.50136799669702123180560377799

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