Properties

Label 2-2475-1.1-c1-0-11
Degree $2$
Conductor $2475$
Sign $1$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 7-s + 11-s − 13-s + 4·16-s + 6·17-s − 7·19-s − 6·23-s + 2·28-s + 6·29-s − 7·31-s + 2·37-s + 6·41-s − 43-s − 2·44-s − 6·49-s + 2·52-s + 6·53-s + 5·61-s − 8·64-s + 5·67-s − 12·68-s + 12·71-s + 14·73-s + 14·76-s − 77-s − 4·79-s + ⋯
L(s)  = 1  − 4-s − 0.377·7-s + 0.301·11-s − 0.277·13-s + 16-s + 1.45·17-s − 1.60·19-s − 1.25·23-s + 0.377·28-s + 1.11·29-s − 1.25·31-s + 0.328·37-s + 0.937·41-s − 0.152·43-s − 0.301·44-s − 6/7·49-s + 0.277·52-s + 0.824·53-s + 0.640·61-s − 64-s + 0.610·67-s − 1.45·68-s + 1.42·71-s + 1.63·73-s + 1.60·76-s − 0.113·77-s − 0.450·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2475} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.117963749\)
\(L(\frac12)\) \(\approx\) \(1.117963749\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.899119495915130256624258150613, −8.240178529480385739684701459320, −7.59106525614709698283393492786, −6.48339477315215306115657995580, −5.80741859506924874487989702702, −4.93966973271202037240525713331, −4.06083776331402365524867624470, −3.42870909379007946401799132003, −2.11432709634023786835252102513, −0.67233943724094663765995022815, 0.67233943724094663765995022815, 2.11432709634023786835252102513, 3.42870909379007946401799132003, 4.06083776331402365524867624470, 4.93966973271202037240525713331, 5.80741859506924874487989702702, 6.48339477315215306115657995580, 7.59106525614709698283393492786, 8.240178529480385739684701459320, 8.899119495915130256624258150613

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