Properties

Label 2-2898-1.1-c1-0-25
Degree $2$
Conductor $2898$
Sign $1$
Analytic cond. $23.1406$
Root an. cond. $4.81047$
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  + 2-s + 4-s + 3.23·5-s − 7-s + 8-s + 3.23·10-s − 6.47·13-s − 14-s + 16-s + 2.76·17-s + 4.47·19-s + 3.23·20-s + 23-s + 5.47·25-s − 6.47·26-s − 28-s + 2·29-s + 3.23·31-s + 32-s + 2.76·34-s − 3.23·35-s + 6·37-s + 4.47·38-s + 3.23·40-s + 10·41-s + 2.47·43-s + 46-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.44·5-s − 0.377·7-s + 0.353·8-s + 1.02·10-s − 1.79·13-s − 0.267·14-s + 0.250·16-s + 0.670·17-s + 1.02·19-s + 0.723·20-s + 0.208·23-s + 1.09·25-s − 1.26·26-s − 0.188·28-s + 0.371·29-s + 0.581·31-s + 0.176·32-s + 0.474·34-s − 0.546·35-s + 0.986·37-s + 0.725·38-s + 0.511·40-s + 1.56·41-s + 0.376·43-s + 0.147·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2898\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(23.1406\)
Root analytic conductor: \(4.81047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2898,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.711629389\)
\(L(\frac12)\) \(\approx\) \(3.711629389\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good5 \( 1 - 3.23T + 5T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 6.47T + 13T^{2} \)
17 \( 1 - 2.76T + 17T^{2} \)
19 \( 1 - 4.47T + 19T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 3.23T + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 2.47T + 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 6.76T + 59T^{2} \)
61 \( 1 + 1.70T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 6.47T + 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 + 6.18T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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

−9.011232051352668053479860286714, −7.69866297603610462479234142356, −7.20095061210415025914491298722, −6.26682397318652874660922930294, −5.62739557677466252773303188339, −5.07368544097290215752507173254, −4.14663407077729232926573451558, −2.77455627253228564364406023046, −2.49038445678904597617129358442, −1.13016948224013605500194388943, 1.13016948224013605500194388943, 2.49038445678904597617129358442, 2.77455627253228564364406023046, 4.14663407077729232926573451558, 5.07368544097290215752507173254, 5.62739557677466252773303188339, 6.26682397318652874660922930294, 7.20095061210415025914491298722, 7.69866297603610462479234142356, 9.011232051352668053479860286714

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