↑

Properties

Label 2-2205-5.4-c1-0-26
Degree $2$
Conductor $2205$
Sign $-0.970 + 0.241i$
Analytic cond. $17.6070$
Root an. cond. $4.19607$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.70i·2-s − 5.34·4-s + (2.17 − 0.539i)5-s − 9.04i·8-s + (1.46 + 5.87i)10-s − 2·11-s + 0.921i·13-s + 13.8·16-s + 1.07i·17-s + 3.07·19-s + (−11.5 + 2.87i)20-s − 5.41i·22-s + 2.34i·23-s + (4.41 − 2.34i)25-s − 2.49·26-s + ⋯
L(s)  = 1  + 1.91i·2-s − 2.67·4-s + (0.970 − 0.241i)5-s − 3.19i·8-s + (0.461 + 1.85i)10-s − 0.603·11-s + 0.255i·13-s + 3.45·16-s + 0.261i·17-s + 0.706·19-s + (−2.59 + 0.643i)20-s − 1.15i·22-s + 0.487i·23-s + (0.883 − 0.468i)25-s − 0.489·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.970 + 0.241i$
Analytic conductor: \(17.6070\)
Root analytic conductor: \(4.19607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :1/2),\ -0.970 + 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.445032416\)
\(L(\frac12)\) \(\approx\) \(1.445032416\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 0.539i)T \)
7 \( 1 \)
good2 \( 1 - 2.70iT - 2T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 0.921iT - 13T^{2} \)
17 \( 1 - 1.07iT - 17T^{2} \)
19 \( 1 - 3.07T + 19T^{2} \)
23 \( 1 - 2.34iT - 23T^{2} \)
29 \( 1 + 6.68T + 29T^{2} \)
31 \( 1 - 7.75T + 31T^{2} \)
37 \( 1 - 10.8iT - 37T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 - 6.52iT - 43T^{2} \)
47 \( 1 - 4.68iT - 47T^{2} \)
53 \( 1 - 3.75iT - 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 4.15T + 61T^{2} \)
67 \( 1 - 4.68iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 7.07iT - 73T^{2} \)
79 \( 1 + 6.15T + 79T^{2} \)
83 \( 1 + 6.83iT - 83T^{2} \)
89 \( 1 + 8.34T + 89T^{2} \)
97 \( 1 - 8.43iT - 97T^{2} \)
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

−9.387509458724433613477892492084, −8.447684142975010449749385034265, −7.85680425389959540558869070459, −7.09361789578612209256183694802, −6.24446035015574646704715732003, −5.76171205159086367588876404526, −4.99072062952936892057256174608, −4.32080834356785249897568255854, −2.96411714493426832030478141394, −1.22215213643024866436128561551, 0.56550848713064643420962290903, 1.82655398677392774121832209764, 2.56547696483063511763760074006, 3.31620368900512514513234200802, 4.35342739496451027615910363289, 5.27234852272548785067541696545, 5.83798104231212307709343358734, 7.24131397809029083386102488164, 8.229541061573364636728075844367, 9.106249191593358257131836845120

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