Properties

Label 2-75e2-1.1-c1-0-121
Degree $2$
Conductor $5625$
Sign $-1$
Analytic cond. $44.9158$
Root an. cond. $6.70192$
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  + 0.618·2-s − 1.61·4-s − 2.85·7-s − 2.23·8-s + 1.61·11-s + 3.47·13-s − 1.76·14-s + 1.85·16-s − 7.47·17-s + 5.85·19-s + 1.00·22-s − 4.85·23-s + 2.14·26-s + 4.61·28-s + 7.23·29-s + 2·31-s + 5.61·32-s − 4.61·34-s + 3·37-s + 3.61·38-s + 2.47·41-s + 8.47·43-s − 2.61·44-s − 3.00·46-s − 9.38·47-s + 1.14·49-s − 5.61·52-s + ⋯
L(s)  = 1  + 0.437·2-s − 0.809·4-s − 1.07·7-s − 0.790·8-s + 0.487·11-s + 0.962·13-s − 0.471·14-s + 0.463·16-s − 1.81·17-s + 1.34·19-s + 0.213·22-s − 1.01·23-s + 0.420·26-s + 0.872·28-s + 1.34·29-s + 0.359·31-s + 0.993·32-s − 0.791·34-s + 0.493·37-s + 0.586·38-s + 0.386·41-s + 1.29·43-s − 0.394·44-s − 0.442·46-s − 1.36·47-s + 0.163·49-s − 0.779·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5625\)    =    \(3^{2} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(44.9158\)
Root analytic conductor: \(6.70192\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5625,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 0.618T + 2T^{2} \)
7 \( 1 + 2.85T + 7T^{2} \)
11 \( 1 - 1.61T + 11T^{2} \)
13 \( 1 - 3.47T + 13T^{2} \)
17 \( 1 + 7.47T + 17T^{2} \)
19 \( 1 - 5.85T + 19T^{2} \)
23 \( 1 + 4.85T + 23T^{2} \)
29 \( 1 - 7.23T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 - 2.47T + 41T^{2} \)
43 \( 1 - 8.47T + 43T^{2} \)
47 \( 1 + 9.38T + 47T^{2} \)
53 \( 1 - 0.472T + 53T^{2} \)
59 \( 1 + 3.94T + 59T^{2} \)
61 \( 1 + 2.14T + 61T^{2} \)
67 \( 1 + 2.52T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + 6.85T + 73T^{2} \)
79 \( 1 - 6.70T + 79T^{2} \)
83 \( 1 - 4.94T + 83T^{2} \)
89 \( 1 + 1.38T + 89T^{2} \)
97 \( 1 + 8.38T + 97T^{2} \)
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

−7.88853698352647366217674120195, −6.79827079079622531146088883879, −6.26546482847243600976778341612, −5.75022788251178262451060567220, −4.65565157931635942122052662935, −4.15187376658563593192546674036, −3.36921134534008706796559270764, −2.65776609986496083070525828405, −1.18586945317934901529615323259, 0, 1.18586945317934901529615323259, 2.65776609986496083070525828405, 3.36921134534008706796559270764, 4.15187376658563593192546674036, 4.65565157931635942122052662935, 5.75022788251178262451060567220, 6.26546482847243600976778341612, 6.79827079079622531146088883879, 7.88853698352647366217674120195

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