Properties

Label 2-1155-33.32-c1-0-25
Degree $2$
Conductor $1155$
Sign $-0.490 - 0.871i$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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  + 0.164·2-s + (1.34 + 1.08i)3-s − 1.97·4-s + i·5-s + (0.221 + 0.179i)6-s i·7-s − 0.654·8-s + (0.630 + 2.93i)9-s + 0.164i·10-s + (3.08 + 1.22i)11-s + (−2.65 − 2.14i)12-s − 1.60i·13-s − 0.164i·14-s + (−1.08 + 1.34i)15-s + 3.83·16-s − 5.06·17-s + ⋯
L(s)  = 1  + 0.116·2-s + (0.777 + 0.628i)3-s − 0.986·4-s + 0.447i·5-s + (0.0906 + 0.0732i)6-s − 0.377i·7-s − 0.231·8-s + (0.210 + 0.977i)9-s + 0.0520i·10-s + (0.929 + 0.369i)11-s + (−0.767 − 0.619i)12-s − 0.445i·13-s − 0.0440i·14-s + (−0.281 + 0.347i)15-s + 0.959·16-s − 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.490 - 0.871i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.490 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.492524573\)
\(L(\frac12)\) \(\approx\) \(1.492524573\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.34 - 1.08i)T \)
5 \( 1 - iT \)
7 \( 1 + iT \)
11 \( 1 + (-3.08 - 1.22i)T \)
good2 \( 1 - 0.164T + 2T^{2} \)
13 \( 1 + 1.60iT - 13T^{2} \)
17 \( 1 + 5.06T + 17T^{2} \)
19 \( 1 - 5.56iT - 19T^{2} \)
23 \( 1 - 6.91iT - 23T^{2} \)
29 \( 1 - 9.80T + 29T^{2} \)
31 \( 1 + 5.21T + 31T^{2} \)
37 \( 1 + 11.6T + 37T^{2} \)
41 \( 1 + 3.68T + 41T^{2} \)
43 \( 1 - 0.866iT - 43T^{2} \)
47 \( 1 + 2.37iT - 47T^{2} \)
53 \( 1 - 6.59iT - 53T^{2} \)
59 \( 1 - 0.691iT - 59T^{2} \)
61 \( 1 - 9.81iT - 61T^{2} \)
67 \( 1 - 5.99T + 67T^{2} \)
71 \( 1 - 7.38iT - 71T^{2} \)
73 \( 1 + 16.3iT - 73T^{2} \)
79 \( 1 - 8.79iT - 79T^{2} \)
83 \( 1 + 2.46T + 83T^{2} \)
89 \( 1 + 7.69iT - 89T^{2} \)
97 \( 1 - 1.23T + 97T^{2} \)
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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

−10.11467817540159038588734209740, −9.165638842790093151354836380401, −8.645118715421660061006755891144, −7.75995136139049950856575957828, −6.86566242344108164330572951950, −5.65061143228460469450119158302, −4.66758360386914419388725060324, −3.87619146208522426868171492014, −3.25531822337859046006368635990, −1.68925941468716607972861355343, 0.59624377808551325635302506332, 2.03634702133017058371918393790, 3.27654166533953763487191718574, 4.29508873853266767223720643732, 5.01203139369414061423097484244, 6.41027263061959417970095856150, 6.87813648309557504386426175020, 8.286142874124697792497845247230, 8.839272552846041122288141941100, 9.023718404768070503481571877506

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