Properties

Label 2-2475-1.1-c3-0-170
Degree $2$
Conductor $2475$
Sign $1$
Analytic cond. $146.029$
Root an. cond. $12.0842$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5.42·2-s + 21.4·4-s + 7.69·7-s + 72.8·8-s + 11·11-s − 24.8·13-s + 41.7·14-s + 223.·16-s − 15.9·17-s + 15.1·19-s + 59.6·22-s + 17.7·23-s − 134.·26-s + 164.·28-s + 128.·29-s + 219.·31-s + 630.·32-s − 86.4·34-s − 92.0·37-s + 82.1·38-s + 459.·41-s − 64.9·43-s + 235.·44-s + 96.3·46-s + 497.·47-s − 283.·49-s − 532.·52-s + ⋯
L(s)  = 1  + 1.91·2-s + 2.67·4-s + 0.415·7-s + 3.21·8-s + 0.301·11-s − 0.530·13-s + 0.797·14-s + 3.49·16-s − 0.227·17-s + 0.182·19-s + 0.578·22-s + 0.160·23-s − 1.01·26-s + 1.11·28-s + 0.823·29-s + 1.27·31-s + 3.48·32-s − 0.436·34-s − 0.409·37-s + 0.350·38-s + 1.75·41-s − 0.230·43-s + 0.807·44-s + 0.308·46-s + 1.54·47-s − 0.827·49-s − 1.41·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(146.029\)
Root analytic conductor: \(12.0842\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(10.08522948\)
\(L(\frac12)\) \(\approx\) \(10.08522948\)
\(L(\frac{5}{2})\) 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 \( 1 - 11T \)
good2 \( 1 - 5.42T + 8T^{2} \)
7 \( 1 - 7.69T + 343T^{2} \)
13 \( 1 + 24.8T + 2.19e3T^{2} \)
17 \( 1 + 15.9T + 4.91e3T^{2} \)
19 \( 1 - 15.1T + 6.85e3T^{2} \)
23 \( 1 - 17.7T + 1.21e4T^{2} \)
29 \( 1 - 128.T + 2.43e4T^{2} \)
31 \( 1 - 219.T + 2.97e4T^{2} \)
37 \( 1 + 92.0T + 5.06e4T^{2} \)
41 \( 1 - 459.T + 6.89e4T^{2} \)
43 \( 1 + 64.9T + 7.95e4T^{2} \)
47 \( 1 - 497.T + 1.03e5T^{2} \)
53 \( 1 + 526.T + 1.48e5T^{2} \)
59 \( 1 - 578.T + 2.05e5T^{2} \)
61 \( 1 + 221.T + 2.26e5T^{2} \)
67 \( 1 - 860.T + 3.00e5T^{2} \)
71 \( 1 + 580.T + 3.57e5T^{2} \)
73 \( 1 + 510.T + 3.89e5T^{2} \)
79 \( 1 - 1.03e3T + 4.93e5T^{2} \)
83 \( 1 - 606.T + 5.71e5T^{2} \)
89 \( 1 - 23.4T + 7.04e5T^{2} \)
97 \( 1 + 719.T + 9.12e5T^{2} \)
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

−8.261579963786393068971411877193, −7.51235855780690021742990014837, −6.73778581700204188735748099078, −6.11976253167072302098022074699, −5.24107771260028231703855754604, −4.62111246498442470999945091099, −3.95696909420329897397791955117, −2.93943365083729515510651542215, −2.25404598355844444123514292016, −1.11018526135846204203044094696, 1.11018526135846204203044094696, 2.25404598355844444123514292016, 2.93943365083729515510651542215, 3.95696909420329897397791955117, 4.62111246498442470999945091099, 5.24107771260028231703855754604, 6.11976253167072302098022074699, 6.73778581700204188735748099078, 7.51235855780690021742990014837, 8.261579963786393068971411877193

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