Properties

Label 2-1155-33.32-c1-0-22
Degree $2$
Conductor $1155$
Sign $-0.356 - 0.934i$
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  − 1.78·2-s + (−1.68 − 0.395i)3-s + 1.17·4-s + i·5-s + (3.00 + 0.704i)6-s + i·7-s + 1.47·8-s + (2.68 + 1.33i)9-s − 1.78i·10-s + (1.85 + 2.74i)11-s + (−1.98 − 0.464i)12-s + 5.68i·13-s − 1.78i·14-s + (0.395 − 1.68i)15-s − 4.96·16-s + 2.05·17-s + ⋯
L(s)  = 1  − 1.25·2-s + (−0.973 − 0.228i)3-s + 0.587·4-s + 0.447i·5-s + (1.22 + 0.287i)6-s + 0.377i·7-s + 0.519·8-s + (0.895 + 0.444i)9-s − 0.563i·10-s + (0.560 + 0.828i)11-s + (−0.571 − 0.134i)12-s + 1.57i·13-s − 0.476i·14-s + (0.102 − 0.435i)15-s − 1.24·16-s + 0.497·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.356 - 0.934i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.356 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5193182926\)
\(L(\frac12)\) \(\approx\) \(0.5193182926\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 + 0.395i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
11 \( 1 + (-1.85 - 2.74i)T \)
good2 \( 1 + 1.78T + 2T^{2} \)
13 \( 1 - 5.68iT - 13T^{2} \)
17 \( 1 - 2.05T + 17T^{2} \)
19 \( 1 - 4.08iT - 19T^{2} \)
23 \( 1 + 5.58iT - 23T^{2} \)
29 \( 1 - 6.60T + 29T^{2} \)
31 \( 1 - 8.58T + 31T^{2} \)
37 \( 1 + 9.54T + 37T^{2} \)
41 \( 1 - 4.92T + 41T^{2} \)
43 \( 1 - 1.16iT - 43T^{2} \)
47 \( 1 - 1.41iT - 47T^{2} \)
53 \( 1 + 4.49iT - 53T^{2} \)
59 \( 1 + 3.29iT - 59T^{2} \)
61 \( 1 - 2.93iT - 61T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 - 7.13iT - 71T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 + 8.84iT - 79T^{2} \)
83 \( 1 - 4.64T + 83T^{2} \)
89 \( 1 - 0.807iT - 89T^{2} \)
97 \( 1 - 7.68T + 97T^{2} \)
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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

−10.19757945891508275301011817226, −9.317898646176613996919212555179, −8.490784766472952355871476961706, −7.57487042085837094203577791926, −6.72260493270247550093370712726, −6.33088831407104428927643435424, −4.87178955720076249360582475068, −4.15660865513300648731494000460, −2.21986913199587006799261886965, −1.25424303114596663457575524572, 0.51934862862717031433487058493, 1.22464316035336197747397318980, 3.23575286723762747661565923476, 4.49639321220900139998563621538, 5.33502546807120925293900208681, 6.26878474226213886920622257077, 7.25071866620270606314133393503, 8.018427893247267051966053184743, 8.809568545334597325557516936877, 9.610598116507336246154785121932

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