Properties

Label 2-2205-315.284-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: $\chi_{2205} (2174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ 0.220 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.381076063\)
\(L(\frac12)\) \(\approx\) \(1.381076063\)
\(L(1)\) 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

−9.179291040763371791126752603730, −8.775422706104966744135415091549, −8.161810151030676458203986576107, −6.70782461752361003388159657240, −6.17050651741043198359451326295, −5.75066415523741377470466000794, −4.44865694576175685041412409581, −3.63234749344303478632032259890, −2.94312829310744692393409048619, −1.53415641740609884891509966318, 1.17557768724919395806095352179, 1.95545074615478379152446187518, 2.78930609763453276351320689692, 4.56619330363123953541590790927, 5.41827720841477317523960941954, 5.89895402691265616349141089682, 6.82891119589023998683851487640, 7.08843877231934599915486727115, 8.246689150730527971556058386911, 9.187286126930038455213650910060

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