Properties

Label 2-1155-5.4-c1-0-6
Degree $2$
Conductor $1155$
Sign $0.973 + 0.227i$
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  − 2.51i·2-s + i·3-s − 4.34·4-s + (−2.17 − 0.508i)5-s + 2.51·6-s i·7-s + 5.89i·8-s − 9-s + (−1.28 + 5.48i)10-s − 11-s − 4.34i·12-s − 1.93i·13-s − 2.51·14-s + (0.508 − 2.17i)15-s + 6.17·16-s + 2.73i·17-s + ⋯
L(s)  = 1  − 1.78i·2-s + 0.577i·3-s − 2.17·4-s + (−0.973 − 0.227i)5-s + 1.02·6-s − 0.377i·7-s + 2.08i·8-s − 0.333·9-s + (−0.405 + 1.73i)10-s − 0.301·11-s − 1.25i·12-s − 0.535i·13-s − 0.673·14-s + (0.131 − 0.562i)15-s + 1.54·16-s + 0.663i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.973 + 0.227i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.973 + 0.227i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5946549721\)
\(L(\frac12)\) \(\approx\) \(0.5946549721\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 + 0.508i)T \)
7 \( 1 + iT \)
11 \( 1 + T \)
good2 \( 1 + 2.51iT - 2T^{2} \)
13 \( 1 + 1.93iT - 13T^{2} \)
17 \( 1 - 2.73iT - 17T^{2} \)
19 \( 1 + 5.74T + 19T^{2} \)
23 \( 1 - 3.98iT - 23T^{2} \)
29 \( 1 - 0.897T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 3.09iT - 37T^{2} \)
41 \( 1 - 7.83T + 41T^{2} \)
43 \( 1 - 4.01iT - 43T^{2} \)
47 \( 1 - 3.41iT - 47T^{2} \)
53 \( 1 - 9.17iT - 53T^{2} \)
59 \( 1 + 9.82T + 59T^{2} \)
61 \( 1 - 1.57T + 61T^{2} \)
67 \( 1 + 5.66iT - 67T^{2} \)
71 \( 1 + 5.57T + 71T^{2} \)
73 \( 1 + 9.45iT - 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 - 8.24iT - 83T^{2} \)
89 \( 1 + 9.21T + 89T^{2} \)
97 \( 1 - 15.8iT - 97T^{2} \)
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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

−10.04467310675595027236608942031, −9.222718517006327986156685709608, −8.371419687694270805353132318181, −7.77610188464877293798878816330, −6.20388924963629299890072585719, −4.81271378532524994161340801916, −4.30854087661665480130243262281, −3.48726459859363373476052943617, −2.62425080374941864198241866135, −1.07846594233262000067809236127, 0.31661409569929309541003264125, 2.61054529769220567061059917990, 4.14239615057523951221644260483, 4.80799867775855498992941571878, 5.92913537611718014788407935269, 6.68872552575695969352712745657, 7.17550827805322147277272551168, 8.185811252011892771063112121173, 8.431954205613351910675100777133, 9.352811992492193332404458039341

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