Properties

Label 2-2205-1.1-c3-0-31
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  − 1.65·2-s − 5.25·4-s + 5·5-s + 21.9·8-s − 8.28·10-s − 69.5·11-s + 68.4·13-s + 5.70·16-s − 104.·17-s + 71.8·19-s − 26.2·20-s + 115.·22-s + 101.·23-s + 25·25-s − 113.·26-s + 114.·29-s − 73.6·31-s − 185.·32-s + 172.·34-s − 200.·37-s − 119.·38-s + 109.·40-s − 417.·41-s + 311.·43-s + 365.·44-s − 167.·46-s − 149.·47-s + ⋯
L(s)  = 1  − 0.585·2-s − 0.657·4-s + 0.447·5-s + 0.970·8-s − 0.261·10-s − 1.90·11-s + 1.45·13-s + 0.0890·16-s − 1.48·17-s + 0.868·19-s − 0.293·20-s + 1.11·22-s + 0.915·23-s + 0.200·25-s − 0.854·26-s + 0.734·29-s − 0.426·31-s − 1.02·32-s + 0.871·34-s − 0.892·37-s − 0.508·38-s + 0.433·40-s − 1.58·41-s + 1.10·43-s + 1.25·44-s − 0.536·46-s − 0.464·47-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: \(0\)
Selberg data: \((2,\ 2205,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.027173579\)
\(L(\frac12)\) \(\approx\) \(1.027173579\)
\(L(\frac{5}{2})\) 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 \( 1 + 1.65T + 8T^{2} \)
11 \( 1 + 69.5T + 1.33e3T^{2} \)
13 \( 1 - 68.4T + 2.19e3T^{2} \)
17 \( 1 + 104.T + 4.91e3T^{2} \)
19 \( 1 - 71.8T + 6.85e3T^{2} \)
23 \( 1 - 101.T + 1.21e4T^{2} \)
29 \( 1 - 114.T + 2.43e4T^{2} \)
31 \( 1 + 73.6T + 2.97e4T^{2} \)
37 \( 1 + 200.T + 5.06e4T^{2} \)
41 \( 1 + 417.T + 6.89e4T^{2} \)
43 \( 1 - 311.T + 7.95e4T^{2} \)
47 \( 1 + 149.T + 1.03e5T^{2} \)
53 \( 1 + 271.T + 1.48e5T^{2} \)
59 \( 1 - 518.T + 2.05e5T^{2} \)
61 \( 1 + 219.T + 2.26e5T^{2} \)
67 \( 1 - 80.6T + 3.00e5T^{2} \)
71 \( 1 - 91.0T + 3.57e5T^{2} \)
73 \( 1 - 882.T + 3.89e5T^{2} \)
79 \( 1 - 599.T + 4.93e5T^{2} \)
83 \( 1 + 70.8T + 5.71e5T^{2} \)
89 \( 1 + 802.T + 7.04e5T^{2} \)
97 \( 1 - 145.T + 9.12e5T^{2} \)
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

−8.640757468656464497898828162807, −8.212291563958438094038602850228, −7.30325371393633714733505691216, −6.43408156581506374296800911242, −5.30845347243766107706461890862, −4.95177163639709088807736015613, −3.77412055282495582050677978420, −2.74796657112291585281408525453, −1.61535225736761742405203726089, −0.51523972753236049132633325599, 0.51523972753236049132633325599, 1.61535225736761742405203726089, 2.74796657112291585281408525453, 3.77412055282495582050677978420, 4.95177163639709088807736015613, 5.30845347243766107706461890862, 6.43408156581506374296800911242, 7.30325371393633714733505691216, 8.212291563958438094038602850228, 8.640757468656464497898828162807

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