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Properties

Label 2-2205-315.254-c0-0-0
Degree $2$
Conductor $2205$
Sign $-0.220 - 0.975i$
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  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s − 5-s + (−0.499 + 0.866i)9-s + 1.73i·11-s + 0.999·12-s + (−1.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.499 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)20-s + 25-s − 0.999·27-s + (−1.49 + 0.866i)33-s + (0.499 + 0.866i)36-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)4-s − 5-s + (−0.499 + 0.866i)9-s + 1.73i·11-s + 0.999·12-s + (−1.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.499 − 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)20-s + 25-s − 0.999·27-s + (−1.49 + 0.866i)33-s + (0.499 + 0.866i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ -0.220 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.025172712\)
\(L(\frac12)\) \(\approx\) \(1.025172712\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + T \)
7 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 - 1.73iT - T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.600892642100508435393616108660, −8.894319468382598474213094982222, −7.69276900333386476894245383641, −7.36557614668499570565895469317, −6.44125132666940221281959470532, −5.15566811704490493899886723474, −4.65446611592523400748536399203, −3.94270764928727950883937312902, −2.66663541707616666672849942209, −1.84731837072163120855307778092, 0.63544797899156085992677169393, 2.46262747250754668426227903794, 3.12864070778596780235475223129, 3.68695707159716777513657252409, 5.05076801859648638703039400063, 6.07431956913791977435424424864, 7.07974904201735100154039819914, 7.48547974340594690698520736418, 8.181375333548074172316527127893, 8.582764038297649325515361448529

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