↑

Properties

Label 2-2205-35.19-c0-0-3
Degree $2$
Conductor $2205$
Sign $-0.327 + 0.944i$
Analytic cond. $1.10043$
Root an. cond. $1.04901$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.60 − 0.923i)2-s + (1.20 + 2.09i)4-s + (−0.5 + 0.866i)5-s − 2.61i·8-s + (1.60 − 0.923i)10-s + (−1.20 + 2.09i)16-s + (−0.707 − 1.22i)17-s + (−1.60 − 0.923i)19-s − 2.41·20-s + (0.662 + 0.382i)23-s + (−0.499 − 0.866i)25-s + (0.662 − 0.382i)31-s + (1.60 − 0.923i)32-s + 2.61i·34-s + (1.70 + 2.95i)38-s + ⋯
L(s)  = 1  + (−1.60 − 0.923i)2-s + (1.20 + 2.09i)4-s + (−0.5 + 0.866i)5-s − 2.61i·8-s + (1.60 − 0.923i)10-s + (−1.20 + 2.09i)16-s + (−0.707 − 1.22i)17-s + (−1.60 − 0.923i)19-s − 2.41·20-s + (0.662 + 0.382i)23-s + (−0.499 − 0.866i)25-s + (0.662 − 0.382i)31-s + (1.60 − 0.923i)32-s + 2.61i·34-s + (1.70 + 2.95i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.327 + 0.944i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ -0.327 + 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3394536605\)
\(L(\frac12)\) \(\approx\) \(0.3394536605\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good2 \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.662 + 0.382i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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

−9.013959089793217263348599992434, −8.501390396408478106814543116830, −7.58984057654411536196098020570, −7.01965558369414429229936082877, −6.39365704797612235034871189335, −4.72547421546277666038899355180, −3.68554338003886451333300567166, −2.73323490613507911150753525478, −2.15121686829880750594575792197, −0.44892375963555798033542493738, 1.14319138358507389071201255584, 2.18308086025671793210298947681, 3.96804948100627732284092312271, 4.87226908759852629570710620865, 5.99364682359664248373277052808, 6.43997667403796468117755051220, 7.43520058834797454957134793665, 8.120371951044886481231349848000, 8.671574954687217189730839403727, 9.046337301792254676923553993386

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