Properties

Label 2-2898-1.1-c1-0-30
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 1.37·5-s − 7-s − 8-s + 1.37·10-s − 4·11-s + 1.37·13-s + 14-s + 16-s + 4.74·17-s + 4·19-s − 1.37·20-s + 4·22-s + 23-s − 3.11·25-s − 1.37·26-s − 28-s − 9.37·29-s + 6.74·31-s − 32-s − 4.74·34-s + 1.37·35-s + 2.62·37-s − 4·38-s + 1.37·40-s + 8.11·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.613·5-s − 0.377·7-s − 0.353·8-s + 0.433·10-s − 1.20·11-s + 0.380·13-s + 0.267·14-s + 0.250·16-s + 1.15·17-s + 0.917·19-s − 0.306·20-s + 0.852·22-s + 0.208·23-s − 0.623·25-s − 0.269·26-s − 0.188·28-s − 1.74·29-s + 1.21·31-s − 0.176·32-s − 0.813·34-s + 0.231·35-s + 0.431·37-s − 0.648·38-s + 0.216·40-s + 1.26·41-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: \(1\)
Selberg data: \((2,\ 2898,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 + T \)
3 \( 1 \)
7 \( 1 + T \)
23 \( 1 - T \)
good5 \( 1 + 1.37T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 1.37T + 13T^{2} \)
17 \( 1 - 4.74T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 + 9.37T + 29T^{2} \)
31 \( 1 - 6.74T + 31T^{2} \)
37 \( 1 - 2.62T + 37T^{2} \)
41 \( 1 - 8.11T + 41T^{2} \)
43 \( 1 - 6.11T + 43T^{2} \)
47 \( 1 + 4.62T + 47T^{2} \)
53 \( 1 + 4.74T + 53T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 7.48T + 89T^{2} \)
97 \( 1 + 9.37T + 97T^{2} \)
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

−8.273103891265869413935274601959, −7.56201928784635752452427021030, −7.37333219103131871544848108594, −6.00469954452246217645829278501, −5.56828305232552346323215667902, −4.37793510505903826212730695620, −3.35411761774041151867754606391, −2.66909224231013193317197729225, −1.26750770067846259783786485162, 0, 1.26750770067846259783786485162, 2.66909224231013193317197729225, 3.35411761774041151867754606391, 4.37793510505903826212730695620, 5.56828305232552346323215667902, 6.00469954452246217645829278501, 7.37333219103131871544848108594, 7.56201928784635752452427021030, 8.273103891265869413935274601959

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