Properties

Label 2-3330-37.36-c1-0-64
Degree $2$
Conductor $3330$
Sign $-0.675 - 0.737i$
Analytic cond. $26.5901$
Root an. cond. $5.15656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s i·5-s − 1.30·7-s + i·8-s − 10-s − 0.690·11-s − 2.69i·13-s + 1.30i·14-s + 16-s − 6.90i·17-s + 2.69i·19-s + i·20-s + 0.690i·22-s − 8.90i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.447i·5-s − 0.495·7-s + 0.353i·8-s − 0.316·10-s − 0.208·11-s − 0.746i·13-s + 0.350i·14-s + 0.250·16-s − 1.67i·17-s + 0.617i·19-s + 0.223i·20-s + 0.147i·22-s − 1.85i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3330\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 37\)
Sign: $-0.675 - 0.737i$
Analytic conductor: \(26.5901\)
Root analytic conductor: \(5.15656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3330} (2071, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3330,\ (\ :1/2),\ -0.675 - 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5262145061\)
\(L(\frac12)\) \(\approx\) \(0.5262145061\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + iT \)
37 \( 1 + (-4.10 - 4.48i)T \)
good7 \( 1 + 1.30T + 7T^{2} \)
11 \( 1 + 0.690T + 11T^{2} \)
13 \( 1 + 2.69iT - 13T^{2} \)
17 \( 1 + 6.90iT - 17T^{2} \)
19 \( 1 - 2.69iT - 19T^{2} \)
23 \( 1 + 8.90iT - 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 - 9.59iT - 31T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 8.21iT - 43T^{2} \)
47 \( 1 - 4.61T + 47T^{2} \)
53 \( 1 + 0.690T + 53T^{2} \)
59 \( 1 - 6.21iT - 59T^{2} \)
61 \( 1 + 2iT - 61T^{2} \)
67 \( 1 + 1.23T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 1.59iT - 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + 5.92iT - 89T^{2} \)
97 \( 1 - 3.38iT - 97T^{2} \)
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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

−8.417184315549201249539245463533, −7.47311733751355557569427970530, −6.69595871306119300560412062937, −5.69700045044607079584507267548, −4.97162906455190267155951724630, −4.27466768848111523994091600264, −3.11481221499821211685596745131, −2.63307335500337638059311192294, −1.23588283155390227869120259849, −0.16903558315963790716294049669, 1.57784748886960220896653656684, 2.77953127010587209090532147899, 3.84191628996425138598798106390, 4.37703866835810480092881128896, 5.67622544107701968304139562046, 6.05048519024268534738842772657, 6.84410541771061295283992573817, 7.61151200920662697442673833122, 8.127223085551192063591751930413, 9.194975301818258551153369681973

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