Properties

Label 2-2205-1.1-c3-0-125
Degree $2$
Conductor $2205$
Sign $-1$
Analytic cond. $130.099$
Root an. cond. $11.4061$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.525·2-s − 7.72·4-s − 5·5-s − 8.26·8-s − 2.62·10-s + 48.9·11-s − 40.1·13-s + 57.4·16-s − 59.1·17-s − 129.·19-s + 38.6·20-s + 25.7·22-s + 93.5·23-s + 25·25-s − 21.1·26-s + 117.·29-s + 162.·31-s + 96.3·32-s − 31.0·34-s − 3.72·37-s − 68.0·38-s + 41.3·40-s + 222.·41-s − 62.2·43-s − 378.·44-s + 49.1·46-s + 460.·47-s + ⋯
L(s)  = 1  + 0.185·2-s − 0.965·4-s − 0.447·5-s − 0.365·8-s − 0.0831·10-s + 1.34·11-s − 0.856·13-s + 0.897·16-s − 0.844·17-s − 1.56·19-s + 0.431·20-s + 0.249·22-s + 0.848·23-s + 0.200·25-s − 0.159·26-s + 0.750·29-s + 0.944·31-s + 0.532·32-s − 0.156·34-s − 0.0165·37-s − 0.290·38-s + 0.163·40-s + 0.848·41-s − 0.220·43-s − 1.29·44-s + 0.157·46-s + 1.42·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(130.099\)
Root analytic conductor: \(11.4061\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2205,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 5T \)
7 \( 1 \)
good2 \( 1 - 0.525T + 8T^{2} \)
11 \( 1 - 48.9T + 1.33e3T^{2} \)
13 \( 1 + 40.1T + 2.19e3T^{2} \)
17 \( 1 + 59.1T + 4.91e3T^{2} \)
19 \( 1 + 129.T + 6.85e3T^{2} \)
23 \( 1 - 93.5T + 1.21e4T^{2} \)
29 \( 1 - 117.T + 2.43e4T^{2} \)
31 \( 1 - 162.T + 2.97e4T^{2} \)
37 \( 1 + 3.72T + 5.06e4T^{2} \)
41 \( 1 - 222.T + 6.89e4T^{2} \)
43 \( 1 + 62.2T + 7.95e4T^{2} \)
47 \( 1 - 460.T + 1.03e5T^{2} \)
53 \( 1 + 492.T + 1.48e5T^{2} \)
59 \( 1 + 153.T + 2.05e5T^{2} \)
61 \( 1 + 194.T + 2.26e5T^{2} \)
67 \( 1 - 667.T + 3.00e5T^{2} \)
71 \( 1 + 256.T + 3.57e5T^{2} \)
73 \( 1 + 1.21e3T + 3.89e5T^{2} \)
79 \( 1 - 381.T + 4.93e5T^{2} \)
83 \( 1 - 281.T + 5.71e5T^{2} \)
89 \( 1 - 926.T + 7.04e5T^{2} \)
97 \( 1 - 634.T + 9.12e5T^{2} \)
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

−8.578811626463639305549818458839, −7.57396245793991325665118756050, −6.66267704266119356059109058122, −6.03409193464911363186114834431, −4.66240244380236316200173159825, −4.49498894284600565048400957899, −3.54739137220826520531178891858, −2.43297761510611664611666088334, −1.04016133135528069436359703467, 0, 1.04016133135528069436359703467, 2.43297761510611664611666088334, 3.54739137220826520531178891858, 4.49498894284600565048400957899, 4.66240244380236316200173159825, 6.03409193464911363186114834431, 6.66267704266119356059109058122, 7.57396245793991325665118756050, 8.578811626463639305549818458839

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