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Properties

Label 2-2205-315.124-c0-0-0
Degree $2$
Conductor $2205$
Sign $-0.592 + 0.805i$
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 − 11-s − 0.999·12-s + (−0.5 − 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 + 1.73i)29-s + (−0.5 − 0.866i)33-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s − 5-s + (−0.499 + 0.866i)9-s − 11-s − 0.999·12-s + (−0.5 − 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 + 1.73i)29-s + (−0.5 − 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.592 + 0.805i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (754, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ -0.592 + 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1526202318\)
\(L(\frac12)\) \(\approx\) \(0.1526202318\)
\(L(1)\) 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 \( 1 + T + T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.566671981594095622006622633838, −8.859568537909559257021859357092, −8.238585446417142323135468133399, −7.62313981504224925251360684452, −7.06772490830395878993606070776, −5.27048683863024817261348554411, −4.93516210497033310093924648800, −3.97150994849339384102716452276, −3.21809515910575157188508358647, −2.60676946862025879113476950774, 0.095927526189540525474787946843, 1.65548428536180489242955050933, 2.64682065646636589172963596635, 3.92035110448656786628469104350, 4.58087534062662674520746157327, 5.68998487486169094848447714222, 6.44522424421298641649971969984, 7.32035518829285188418376571392, 7.974274197515447138288387367757, 8.637373265685509589425506990638

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