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Properties

Label 2-2205-15.14-c0-0-3
Degree $2$
Conductor $2205$
Sign $-0.816 + 0.577i$
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  − 4-s − i·5-s − 1.41i·11-s + 16-s − 1.41·19-s + i·20-s − 25-s + 1.41i·29-s − 1.41·31-s − 2i·41-s + 1.41i·44-s − 1.41·55-s − 2i·59-s − 1.41·61-s − 64-s + ⋯
L(s)  = 1  − 4-s − i·5-s − 1.41i·11-s + 16-s − 1.41·19-s + i·20-s − 25-s + 1.41i·29-s − 1.41·31-s − 2i·41-s + 1.41i·44-s − 1.41·55-s − 2i·59-s − 1.41·61-s − 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.816 + 0.577i$
Analytic conductor: \(1.10043\)
Root analytic conductor: \(1.04901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :0),\ -0.816 + 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5444186332\)
\(L(\frac12)\) \(\approx\) \(0.5444186332\)
\(L(1)\) 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 \( 1 + iT \)
7 \( 1 \)
good2 \( 1 + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + 2iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
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

−8.948093649110878637379499019539, −8.404539831691241963369499875731, −7.66120886603086525391819984192, −6.41390674723578661375941948737, −5.55220576182518960643571136279, −5.01729569447184465611958704213, −4.03214868423779175178967646201, −3.38529920852388676523102982087, −1.75300640296577454849462035274, −0.38395689092828449766044622913, 1.83646764645277111645246810794, 2.88514810767459438874630386765, 4.09905637911752007477259236112, 4.47919941536393903385415750521, 5.65674313024701382258900187343, 6.43343922139831240903746898934, 7.32028308940959550528201201929, 7.938097765797023784476776439064, 8.845516638419074934390425149400, 9.681170305292410752120711835702

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