Properties

Label 2-1155-33.32-c1-0-23
Degree $2$
Conductor $1155$
Sign $-0.870 - 0.492i$
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  + 1.41·2-s + (1 + 1.41i)3-s + i·5-s + (1.41 + 2.00i)6-s + i·7-s − 2.82·8-s + (−1.00 + 2.82i)9-s + 1.41i·10-s + (−3 + 1.41i)11-s − 2.24i·13-s + 1.41i·14-s + (−1.41 + i)15-s − 4.00·16-s − 5.82·17-s + (−1.41 + 4.00i)18-s + 3.24i·19-s + ⋯
L(s)  = 1  + 1.00·2-s + (0.577 + 0.816i)3-s + 0.447i·5-s + (0.577 + 0.816i)6-s + 0.377i·7-s − 0.999·8-s + (−0.333 + 0.942i)9-s + 0.447i·10-s + (−0.904 + 0.426i)11-s − 0.621i·13-s + 0.377i·14-s + (−0.365 + 0.258i)15-s − 1.00·16-s − 1.41·17-s + (−0.333 + 0.942i)18-s + 0.743i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.870 - 0.492i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.870 - 0.492i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.892414867\)
\(L(\frac12)\) \(\approx\) \(1.892414867\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1 - 1.41i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
11 \( 1 + (3 - 1.41i)T \)
good2 \( 1 - 1.41T + 2T^{2} \)
13 \( 1 + 2.24iT - 13T^{2} \)
17 \( 1 + 5.82T + 17T^{2} \)
19 \( 1 - 3.24iT - 19T^{2} \)
23 \( 1 - 1.24iT - 23T^{2} \)
29 \( 1 - 5.82T + 29T^{2} \)
31 \( 1 - 8.24T + 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 + 4.58T + 41T^{2} \)
43 \( 1 - 3.24iT - 43T^{2} \)
47 \( 1 - 1.07iT - 47T^{2} \)
53 \( 1 - 4.41iT - 53T^{2} \)
59 \( 1 - 12.8iT - 59T^{2} \)
61 \( 1 + 9.24iT - 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 - 6.48iT - 73T^{2} \)
79 \( 1 - 6.24iT - 79T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 - 16.4iT - 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−10.13192991536889851883323599606, −9.411180763752445079242197060346, −8.475996808158554914037102510333, −7.86203151545594919909184161528, −6.52321052831688718640739036650, −5.66180039340876746857808495419, −4.77517801951387177258266818107, −4.19384811111398399880030679746, −3.00108665063701512194098575358, −2.49604253961629437896164921632, 0.51968838239840333391418286012, 2.28351808249928811457742565149, 3.13364371502824762233945121972, 4.33492508349302071643485766775, 4.91179356007377517280699769106, 6.19853898811194708671325459513, 6.68341426209116621916871477674, 7.83861477618943217929226722440, 8.650414389463137057219876824771, 9.139758022301646199359626877677

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