Properties

Label 2-5e4-1.1-c1-0-20
Degree $2$
Conductor $625$
Sign $-1$
Analytic cond. $4.99065$
Root an. cond. $2.23397$
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  − 1.61·2-s + 3-s + 0.618·4-s − 1.61·6-s + 0.618·7-s + 2.23·8-s − 2·9-s − 5.23·11-s + 0.618·12-s + 1.85·13-s − 1.00·14-s − 4.85·16-s − 5.23·17-s + 3.23·18-s + 0.854·19-s + 0.618·21-s + 8.47·22-s + 3.76·23-s + 2.23·24-s − 3·26-s − 5·27-s + 0.381·28-s − 3.61·29-s − 3·31-s + 3.38·32-s − 5.23·33-s + 8.47·34-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.577·3-s + 0.309·4-s − 0.660·6-s + 0.233·7-s + 0.790·8-s − 0.666·9-s − 1.57·11-s + 0.178·12-s + 0.514·13-s − 0.267·14-s − 1.21·16-s − 1.26·17-s + 0.762·18-s + 0.195·19-s + 0.134·21-s + 1.80·22-s + 0.784·23-s + 0.456·24-s − 0.588·26-s − 0.962·27-s + 0.0721·28-s − 0.671·29-s − 0.538·31-s + 0.597·32-s − 0.911·33-s + 1.45·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $-1$
Analytic conductor: \(4.99065\)
Root analytic conductor: \(2.23397\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 625,\ (\ :1/2),\ -1)\)

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 \( 1 \)
good2 \( 1 + 1.61T + 2T^{2} \)
3 \( 1 - T + 3T^{2} \)
7 \( 1 - 0.618T + 7T^{2} \)
11 \( 1 + 5.23T + 11T^{2} \)
13 \( 1 - 1.85T + 13T^{2} \)
17 \( 1 + 5.23T + 17T^{2} \)
19 \( 1 - 0.854T + 19T^{2} \)
23 \( 1 - 3.76T + 23T^{2} \)
29 \( 1 + 3.61T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 0.236T + 37T^{2} \)
41 \( 1 + 0.763T + 41T^{2} \)
43 \( 1 + 4.85T + 43T^{2} \)
47 \( 1 - 0.618T + 47T^{2} \)
53 \( 1 + 3.47T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 - 8.70T + 61T^{2} \)
67 \( 1 - 4.76T + 67T^{2} \)
71 \( 1 + 6.61T + 71T^{2} \)
73 \( 1 + 9T + 73T^{2} \)
79 \( 1 + 8.09T + 79T^{2} \)
83 \( 1 + 6.23T + 83T^{2} \)
89 \( 1 + 8.94T + 89T^{2} \)
97 \( 1 + 3.85T + 97T^{2} \)
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

−10.01652423904056671200617946383, −9.101642758873869954633728325187, −8.506646481105811914530586321941, −7.88479164562301327349912281171, −7.01893864734912219439462109009, −5.60320073730555610124605744204, −4.58034193435069674182828980796, −3.07879746442872269965969755136, −1.92322418237009403501123881674, 0, 1.92322418237009403501123881674, 3.07879746442872269965969755136, 4.58034193435069674182828980796, 5.60320073730555610124605744204, 7.01893864734912219439462109009, 7.88479164562301327349912281171, 8.506646481105811914530586321941, 9.101642758873869954633728325187, 10.01652423904056671200617946383

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