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Properties

Label 2-1155-33.32-c1-0-94
Degree $2$
Conductor $1155$
Sign $-0.891 - 0.452i$
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  + 0.892·2-s + (0.497 − 1.65i)3-s − 1.20·4-s − i·5-s + (0.444 − 1.48i)6-s + i·7-s − 2.85·8-s + (−2.50 − 1.65i)9-s − 0.892i·10-s + (−0.586 + 3.26i)11-s + (−0.598 + 1.99i)12-s − 5.84i·13-s + 0.892i·14-s + (−1.65 − 0.497i)15-s − 0.147·16-s − 4.41·17-s + ⋯
L(s)  = 1  + 0.631·2-s + (0.287 − 0.957i)3-s − 0.601·4-s − 0.447i·5-s + (0.181 − 0.604i)6-s + 0.377i·7-s − 1.01·8-s + (−0.834 − 0.550i)9-s − 0.282i·10-s + (−0.176 + 0.984i)11-s + (−0.172 + 0.576i)12-s − 1.62i·13-s + 0.238i·14-s + (−0.428 − 0.128i)15-s − 0.0367·16-s − 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.891 - 0.452i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.891 - 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4622960719\)
\(L(\frac12)\) \(\approx\) \(0.4622960719\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.497 + 1.65i)T \)
5 \( 1 + iT \)
7 \( 1 - iT \)
11 \( 1 + (0.586 - 3.26i)T \)
good2 \( 1 - 0.892T + 2T^{2} \)
13 \( 1 + 5.84iT - 13T^{2} \)
17 \( 1 + 4.41T + 17T^{2} \)
19 \( 1 + 1.11iT - 19T^{2} \)
23 \( 1 - 7.41iT - 23T^{2} \)
29 \( 1 + 9.75T + 29T^{2} \)
31 \( 1 + 0.0544T + 31T^{2} \)
37 \( 1 + 4.09T + 37T^{2} \)
41 \( 1 + 0.969T + 41T^{2} \)
43 \( 1 - 0.978iT - 43T^{2} \)
47 \( 1 + 11.0iT - 47T^{2} \)
53 \( 1 - 0.255iT - 53T^{2} \)
59 \( 1 - 1.86iT - 59T^{2} \)
61 \( 1 + 5.07iT - 61T^{2} \)
67 \( 1 - 5.79T + 67T^{2} \)
71 \( 1 - 14.9iT - 71T^{2} \)
73 \( 1 + 7.44iT - 73T^{2} \)
79 \( 1 - 7.49iT - 79T^{2} \)
83 \( 1 + 5.81T + 83T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 - 1.50T + 97T^{2} \)
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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

−9.196772753832818185424230579447, −8.468028991197720227130807049551, −7.69879106006990410408939459902, −6.84447747058614232432073735061, −5.53951374307195275873702358622, −5.35145409322085419932254318341, −3.99335157631283737205607049901, −3.01970109825192498641042600779, −1.83514697298155107928165303902, −0.14965848719044039909712088563, 2.36335033164973788195330332046, 3.51751460993063520660175136982, 4.15643944311542128271260738085, 4.85843675415374271989213313229, 5.92920310180051089715285779052, 6.68637159132805066840076120369, 7.996897801242175332617855833639, 8.958347127402205452325203965513, 9.216007538178822711720201803364, 10.30151850615349989194378378112

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