Properties

Label 2-1155-33.32-c1-0-15
Degree $2$
Conductor $1155$
Sign $-0.902 + 0.431i$
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  − 0.371·2-s + (−0.305 + 1.70i)3-s − 1.86·4-s + i·5-s + (0.113 − 0.634i)6-s + i·7-s + 1.43·8-s + (−2.81 − 1.04i)9-s − 0.371i·10-s + (1.93 + 2.69i)11-s + (0.569 − 3.17i)12-s + 6.33i·13-s − 0.371i·14-s + (−1.70 − 0.305i)15-s + 3.18·16-s − 1.53·17-s + ⋯
L(s)  = 1  − 0.262·2-s + (−0.176 + 0.984i)3-s − 0.930·4-s + 0.447i·5-s + (0.0464 − 0.258i)6-s + 0.377i·7-s + 0.507·8-s + (−0.937 − 0.347i)9-s − 0.117i·10-s + (0.583 + 0.811i)11-s + (0.164 − 0.916i)12-s + 1.75i·13-s − 0.0993i·14-s + (−0.440 − 0.0789i)15-s + 0.797·16-s − 0.373·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.902 + 0.431i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.902 + 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6368188562\)
\(L(\frac12)\) \(\approx\) \(0.6368188562\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.305 - 1.70i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
11 \( 1 + (-1.93 - 2.69i)T \)
good2 \( 1 + 0.371T + 2T^{2} \)
13 \( 1 - 6.33iT - 13T^{2} \)
17 \( 1 + 1.53T + 17T^{2} \)
19 \( 1 - 1.59iT - 19T^{2} \)
23 \( 1 - 0.203iT - 23T^{2} \)
29 \( 1 - 2.40T + 29T^{2} \)
31 \( 1 + 5.91T + 31T^{2} \)
37 \( 1 - 8.80T + 37T^{2} \)
41 \( 1 + 9.29T + 41T^{2} \)
43 \( 1 - 9.25iT - 43T^{2} \)
47 \( 1 + 4.56iT - 47T^{2} \)
53 \( 1 - 2.51iT - 53T^{2} \)
59 \( 1 + 4.95iT - 59T^{2} \)
61 \( 1 + 8.69iT - 61T^{2} \)
67 \( 1 + 4.79T + 67T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + 7.53iT - 73T^{2} \)
79 \( 1 - 6.74iT - 79T^{2} \)
83 \( 1 + 5.23T + 83T^{2} \)
89 \( 1 + 10.6iT - 89T^{2} \)
97 \( 1 - 4.65T + 97T^{2} \)
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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

−9.970726026364870240844900597031, −9.430782583281481507108884534012, −8.989315931449803671516068522759, −8.042105454343546140357633819332, −6.85821991608430021578965006799, −6.03742152409132110050436268590, −4.82553312587331708213446619834, −4.33647903027873896634213854576, −3.42893599188486521950773125605, −1.87111116200563631591649177483, 0.36215754128752573331343067612, 1.22885042426074695084266069964, 2.93115745147043760400496546569, 4.04426858764199935320007512205, 5.26309995245388313207722687020, 5.80854060460704125422162777900, 6.97007979232507469018418652779, 7.86851460162197924786502245018, 8.455493591438469318083984935883, 9.063195345940924893090988079174

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