Properties

Label 2-2475-1.1-c1-0-30
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.30·2-s + 3.30·4-s + 4.30·7-s − 3.00·8-s + 11-s + 5·13-s − 9.90·14-s + 0.302·16-s + 3.90·17-s − 19-s − 2.30·22-s + 3.69·23-s − 11.5·26-s + 14.2·28-s + 9.90·29-s − 4.21·31-s + 5.30·32-s − 9·34-s + 9.60·37-s + 2.30·38-s − 1.60·41-s − 7.21·43-s + 3.30·44-s − 8.51·46-s + 3·47-s + 11.5·49-s + 16.5·52-s + ⋯
L(s)  = 1  − 1.62·2-s + 1.65·4-s + 1.62·7-s − 1.06·8-s + 0.301·11-s + 1.38·13-s − 2.64·14-s + 0.0756·16-s + 0.947·17-s − 0.229·19-s − 0.490·22-s + 0.770·23-s − 2.25·26-s + 2.68·28-s + 1.83·29-s − 0.756·31-s + 0.937·32-s − 1.54·34-s + 1.57·37-s + 0.373·38-s − 0.250·41-s − 1.09·43-s + 0.497·44-s − 1.25·46-s + 0.437·47-s + 1.64·49-s + 2.29·52-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

\(L(1)\) \(\approx\) \(1.223346171\)
\(L(\frac12)\) \(\approx\) \(1.223346171\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.30T + 2T^{2} \)
7 \( 1 - 4.30T + 7T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 - 3.90T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 3.69T + 23T^{2} \)
29 \( 1 - 9.90T + 29T^{2} \)
31 \( 1 + 4.21T + 31T^{2} \)
37 \( 1 - 9.60T + 37T^{2} \)
41 \( 1 + 1.60T + 41T^{2} \)
43 \( 1 + 7.21T + 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 2.30T + 53T^{2} \)
59 \( 1 + 0.211T + 59T^{2} \)
61 \( 1 - 2.90T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 4.60T + 71T^{2} \)
73 \( 1 - 2.90T + 73T^{2} \)
79 \( 1 + 0.0916T + 79T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + 5.30T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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

−8.700451983004077040091605914921, −8.328005907094684800681941219256, −7.77052298978826926256323429642, −6.93028653700106942772943223414, −6.07991306065274212854240727357, −5.05452977537316869129938134035, −4.11494686667864349522574341279, −2.78632753050404375283747449689, −1.52662039952261864760430324013, −1.05921578965706264750828902003, 1.05921578965706264750828902003, 1.52662039952261864760430324013, 2.78632753050404375283747449689, 4.11494686667864349522574341279, 5.05452977537316869129938134035, 6.07991306065274212854240727357, 6.93028653700106942772943223414, 7.77052298978826926256323429642, 8.328005907094684800681941219256, 8.700451983004077040091605914921

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