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Properties

Label 2-2205-35.19-c0-0-5
Degree $2$
Conductor $2205$
Sign $-0.769 + 0.638i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.662 − 0.382i)2-s + (−0.207 − 0.358i)4-s + (0.5 − 0.866i)5-s + 1.08i·8-s + (−0.662 + 0.382i)10-s + (0.207 − 0.358i)16-s + (−0.707 − 1.22i)17-s + (0.662 + 0.382i)19-s − 0.414·20-s + (−1.60 − 0.923i)23-s + (−0.499 − 0.866i)25-s + (1.60 − 0.923i)31-s + (0.662 − 0.382i)32-s + 1.08i·34-s + (−0.292 − 0.507i)38-s + ⋯
L(s)  = 1  + (−0.662 − 0.382i)2-s + (−0.207 − 0.358i)4-s + (0.5 − 0.866i)5-s + 1.08i·8-s + (−0.662 + 0.382i)10-s + (0.207 − 0.358i)16-s + (−0.707 − 1.22i)17-s + (0.662 + 0.382i)19-s − 0.414·20-s + (−1.60 − 0.923i)23-s + (−0.499 − 0.866i)25-s + (1.60 − 0.923i)31-s + (0.662 − 0.382i)32-s + 1.08i·34-s + (−0.292 − 0.507i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.769 + 0.638i$
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: \((2,\ 2205,\ (\ :0),\ -0.769 + 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6821392427\)
\(L(\frac12)\) \(\approx\) \(0.6821392427\)
\(L(1)\) 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 \( 1 + (0.662 + 0.382i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.60 + 0.923i)T + (0.5 + 0.866i)T^{2} \)
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 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.120458657728053086602082861600, −8.327090278232257100801121700661, −7.76651899447604672834164359310, −6.44593303538760019544952394676, −5.77483678973653774268805200461, −4.88853792001856942855320364422, −4.29256853611815424178839330599, −2.67730847793119111671440627344, −1.82232541829262420709486264136, −0.60254194696951926113650233472, 1.62137328669047623432903397718, 2.90127826531929427171888329201, 3.73478876453561576375956122154, 4.68816336414047060900735119437, 6.04677707766473572325722521782, 6.41822311185207957534671417514, 7.39730303170634793748171954892, 7.928361392451228790141532931652, 8.749740816589229497714828746614, 9.506800340433457290277732072676

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