Properties

Label 2-2475-1.1-c1-0-57
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 $1$

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: \(1\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ -1)\)

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 \( 1 \)
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

−8.768041607540139167824161107174, −8.001893150916943033064179219920, −6.94177064779494456719577763043, −6.25997416322047035967534085706, −5.23058676244714842631111750230, −4.48881609881667067649164980343, −3.93260401970958534385260320255, −2.69203750343984022101198568163, −1.44625488225869633667383305184, 0, 1.44625488225869633667383305184, 2.69203750343984022101198568163, 3.93260401970958534385260320255, 4.48881609881667067649164980343, 5.23058676244714842631111750230, 6.25997416322047035967534085706, 6.94177064779494456719577763043, 8.001893150916943033064179219920, 8.768041607540139167824161107174

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