Properties

Label 2-1155-77.76-c1-0-55
Degree $2$
Conductor $1155$
Sign $-0.984 + 0.176i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.26i·2-s + i·3-s + 0.399·4-s + i·5-s + 1.26·6-s + (−0.0556 − 2.64i)7-s − 3.03i·8-s − 9-s + 1.26·10-s + (−0.516 − 3.27i)11-s + 0.399i·12-s − 6.21·13-s + (−3.34 + 0.0703i)14-s − 15-s − 3.04·16-s − 1.62·17-s + ⋯
L(s)  = 1  − 0.894i·2-s + 0.577i·3-s + 0.199·4-s + 0.447i·5-s + 0.516·6-s + (−0.0210 − 0.999i)7-s − 1.07i·8-s − 0.333·9-s + 0.400·10-s + (−0.155 − 0.987i)11-s + 0.115i·12-s − 1.72·13-s + (−0.894 + 0.0188i)14-s − 0.258·15-s − 0.760·16-s − 0.394·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.984 + 0.176i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.984 + 0.176i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9180170632\)
\(L(\frac12)\) \(\approx\) \(0.9180170632\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 - iT \)
7 \( 1 + (0.0556 + 2.64i)T \)
11 \( 1 + (0.516 + 3.27i)T \)
good2 \( 1 + 1.26iT - 2T^{2} \)
13 \( 1 + 6.21T + 13T^{2} \)
17 \( 1 + 1.62T + 17T^{2} \)
19 \( 1 + 4.38T + 19T^{2} \)
23 \( 1 + 1.24T + 23T^{2} \)
29 \( 1 - 9.91iT - 29T^{2} \)
31 \( 1 + 7.78iT - 31T^{2} \)
37 \( 1 - 1.61T + 37T^{2} \)
41 \( 1 - 4.95T + 41T^{2} \)
43 \( 1 + 3.35iT - 43T^{2} \)
47 \( 1 + 4.74iT - 47T^{2} \)
53 \( 1 + 8.51T + 53T^{2} \)
59 \( 1 - 4.49iT - 59T^{2} \)
61 \( 1 - 6.43T + 61T^{2} \)
67 \( 1 - 3.96T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 + 8.16T + 73T^{2} \)
79 \( 1 + 13.4iT - 79T^{2} \)
83 \( 1 - 2.78T + 83T^{2} \)
89 \( 1 - 2.92iT - 89T^{2} \)
97 \( 1 + 15.7iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.841622795533875782021726257072, −8.846666552420074172730670364819, −7.65813476365512892500556255238, −6.98132200978374220066870356389, −6.07474118034903881075544305400, −4.78538504407213385991267811303, −3.90557472856821504910825232370, −3.05684232264936008261732432914, −2.11382030136272731361383726755, −0.34979714002141833801342343303, 2.07541577303936242743016012276, 2.53066812387758159948083743397, 4.56716902364069653543320989894, 5.20369907262918504214766824501, 6.18047046990194290971786474010, 6.81654248849165644693015524701, 7.73130280901320768007722618544, 8.206015466107449140557494795178, 9.205470133885255613600115279728, 9.921949553805496543040599331178

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