Properties

Label 2-2475-1.1-c1-0-2
Degree $2$
Conductor $2475$
Sign $1$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.193·2-s − 1.96·4-s − 3.35·7-s + 0.768·8-s − 11-s − 2.96·13-s + 0.649·14-s + 3.77·16-s − 4.57·17-s − 4.31·19-s + 0.193·22-s − 6.70·23-s + 0.574·26-s + 6.57·28-s + 3.61·29-s + 9.92·31-s − 2.26·32-s + 0.887·34-s + 2·37-s + 0.836·38-s + 4.38·41-s + 9.27·43-s + 1.96·44-s + 1.29·46-s − 9.92·47-s + 4.22·49-s + 5.81·52-s + ⋯
L(s)  = 1  − 0.137·2-s − 0.981·4-s − 1.26·7-s + 0.271·8-s − 0.301·11-s − 0.821·13-s + 0.173·14-s + 0.943·16-s − 1.10·17-s − 0.989·19-s + 0.0413·22-s − 1.39·23-s + 0.112·26-s + 1.24·28-s + 0.670·29-s + 1.78·31-s − 0.401·32-s + 0.152·34-s + 0.328·37-s + 0.135·38-s + 0.685·41-s + 1.41·43-s + 0.295·44-s + 0.191·46-s − 1.44·47-s + 0.603·49-s + 0.806·52-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: no
Arithmetic: yes
Character: $\chi_{2475} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5414767902\)
\(L(\frac12)\) \(\approx\) \(0.5414767902\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.193T + 2T^{2} \)
7 \( 1 + 3.35T + 7T^{2} \)
13 \( 1 + 2.96T + 13T^{2} \)
17 \( 1 + 4.57T + 17T^{2} \)
19 \( 1 + 4.31T + 19T^{2} \)
23 \( 1 + 6.70T + 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
31 \( 1 - 9.92T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 4.38T + 41T^{2} \)
43 \( 1 - 9.27T + 43T^{2} \)
47 \( 1 + 9.92T + 47T^{2} \)
53 \( 1 - 4.70T + 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 8.70T + 61T^{2} \)
67 \( 1 + 5.92T + 67T^{2} \)
71 \( 1 + 9.92T + 71T^{2} \)
73 \( 1 - 7.73T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 - 2.77T + 89T^{2} \)
97 \( 1 + 0.0752T + 97T^{2} \)
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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

−9.025897326995239827070345955448, −8.249279634338735514916731155355, −7.54487065004958476052937792256, −6.40665978546195629469866990454, −6.04825957625956048759198549546, −4.69813017101905642134716650735, −4.32036737446344089561028567884, −3.19152721554331398219805452977, −2.24929861665857050811999273130, −0.45525087172520054253600267706, 0.45525087172520054253600267706, 2.24929861665857050811999273130, 3.19152721554331398219805452977, 4.32036737446344089561028567884, 4.69813017101905642134716650735, 6.04825957625956048759198549546, 6.40665978546195629469866990454, 7.54487065004958476052937792256, 8.249279634338735514916731155355, 9.025897326995239827070345955448

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