Properties

Label 2-2475-1.1-c1-0-17
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  − 0.414·2-s − 1.82·4-s + 2.41·7-s + 1.58·8-s + 11-s − 2.82·13-s − 0.999·14-s + 3·16-s − 0.414·17-s + 3.58·19-s − 0.414·22-s + 23-s + 1.17·26-s − 4.41·28-s − 6.82·29-s + 8.48·31-s − 4.41·32-s + 0.171·34-s + 5.82·37-s − 1.48·38-s − 8.89·41-s + 0.343·43-s − 1.82·44-s − 0.414·46-s + 9.48·47-s − 1.17·49-s + 5.17·52-s + ⋯
L(s)  = 1  − 0.292·2-s − 0.914·4-s + 0.912·7-s + 0.560·8-s + 0.301·11-s − 0.784·13-s − 0.267·14-s + 0.750·16-s − 0.100·17-s + 0.822·19-s − 0.0883·22-s + 0.208·23-s + 0.229·26-s − 0.834·28-s − 1.26·29-s + 1.52·31-s − 0.780·32-s + 0.0294·34-s + 0.958·37-s − 0.240·38-s − 1.38·41-s + 0.0523·43-s − 0.275·44-s − 0.0610·46-s + 1.38·47-s − 0.167·49-s + 0.717·52-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

\(L(1)\) \(\approx\) \(1.324931940\)
\(L(\frac12)\) \(\approx\) \(1.324931940\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.414T + 2T^{2} \)
7 \( 1 - 2.41T + 7T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + 0.414T + 17T^{2} \)
19 \( 1 - 3.58T + 19T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 + 6.82T + 29T^{2} \)
31 \( 1 - 8.48T + 31T^{2} \)
37 \( 1 - 5.82T + 37T^{2} \)
41 \( 1 + 8.89T + 41T^{2} \)
43 \( 1 - 0.343T + 43T^{2} \)
47 \( 1 - 9.48T + 47T^{2} \)
53 \( 1 + 3.65T + 53T^{2} \)
59 \( 1 + 11T + 59T^{2} \)
61 \( 1 - 3.17T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 2.17T + 71T^{2} \)
73 \( 1 - 3.17T + 73T^{2} \)
79 \( 1 - 4.75T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 - 7.65T + 89T^{2} \)
97 \( 1 + 0.171T + 97T^{2} \)
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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

−9.038303246615674129741912420181, −8.047789792445424609641471308052, −7.74193408000098379774415786049, −6.74719232777622347283862389862, −5.60139774587708668935289470370, −4.91449834580153901885306396802, −4.30585944351200599425917937624, −3.25409295182062201412747301077, −1.94093501066343588614189941391, −0.798073002033207128150251416184, 0.798073002033207128150251416184, 1.94093501066343588614189941391, 3.25409295182062201412747301077, 4.30585944351200599425917937624, 4.91449834580153901885306396802, 5.60139774587708668935289470370, 6.74719232777622347283862389862, 7.74193408000098379774415786049, 8.047789792445424609641471308052, 9.038303246615674129741912420181

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