Properties

Label 2-2205-1.1-c3-0-5
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.31·2-s − 2.63·4-s − 5·5-s − 24.6·8-s − 11.5·10-s − 46.2·11-s − 61.3·13-s − 36·16-s − 101.·17-s + 3.66·19-s + 13.1·20-s − 107.·22-s − 84.8·23-s + 25·25-s − 142.·26-s − 30.1·29-s + 188.·31-s + 113.·32-s − 234.·34-s + 18.0·37-s + 8.49·38-s + 123.·40-s − 481.·41-s − 97.7·43-s + 121.·44-s − 196.·46-s − 117.·47-s + ⋯
L(s)  = 1  + 0.819·2-s − 0.329·4-s − 0.447·5-s − 1.08·8-s − 0.366·10-s − 1.26·11-s − 1.30·13-s − 0.562·16-s − 1.44·17-s + 0.0442·19-s + 0.147·20-s − 1.03·22-s − 0.769·23-s + 0.200·25-s − 1.07·26-s − 0.193·29-s + 1.09·31-s + 0.627·32-s − 1.18·34-s + 0.0802·37-s + 0.0362·38-s + 0.486·40-s − 1.83·41-s − 0.346·43-s + 0.417·44-s − 0.630·46-s − 0.365·47-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4874092712\)
\(L(\frac12)\) \(\approx\) \(0.4874092712\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.31T + 8T^{2} \)
11 \( 1 + 46.2T + 1.33e3T^{2} \)
13 \( 1 + 61.3T + 2.19e3T^{2} \)
17 \( 1 + 101.T + 4.91e3T^{2} \)
19 \( 1 - 3.66T + 6.85e3T^{2} \)
23 \( 1 + 84.8T + 1.21e4T^{2} \)
29 \( 1 + 30.1T + 2.43e4T^{2} \)
31 \( 1 - 188.T + 2.97e4T^{2} \)
37 \( 1 - 18.0T + 5.06e4T^{2} \)
41 \( 1 + 481.T + 6.89e4T^{2} \)
43 \( 1 + 97.7T + 7.95e4T^{2} \)
47 \( 1 + 117.T + 1.03e5T^{2} \)
53 \( 1 + 667.T + 1.48e5T^{2} \)
59 \( 1 + 57.3T + 2.05e5T^{2} \)
61 \( 1 - 738.T + 2.26e5T^{2} \)
67 \( 1 - 552.T + 3.00e5T^{2} \)
71 \( 1 - 740.T + 3.57e5T^{2} \)
73 \( 1 - 233.T + 3.89e5T^{2} \)
79 \( 1 + 1.07e3T + 4.93e5T^{2} \)
83 \( 1 + 683.T + 5.71e5T^{2} \)
89 \( 1 - 1.38e3T + 7.04e5T^{2} \)
97 \( 1 - 218.T + 9.12e5T^{2} \)
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

−8.488961836633519815974487581540, −8.052355647212777719591672763677, −7.02344270068642938161531517505, −6.28093293821484073503299374379, −5.11600439857610325225680719294, −4.85891926897435628495308125875, −3.95134973313219901378139161800, −2.94948912226666004718973292477, −2.18604365952443284349527720437, −0.25803942653553343656550071592, 0.25803942653553343656550071592, 2.18604365952443284349527720437, 2.94948912226666004718973292477, 3.95134973313219901378139161800, 4.85891926897435628495308125875, 5.11600439857610325225680719294, 6.28093293821484073503299374379, 7.02344270068642938161531517505, 8.052355647212777719591672763677, 8.488961836633519815974487581540

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