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Properties

Label 2-1155-33.32-c1-0-90
Degree $2$
Conductor $1155$
Sign $-0.830 + 0.556i$
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  + 1.71·2-s + (−0.134 − 1.72i)3-s + 0.925·4-s + i·5-s + (−0.230 − 2.95i)6-s + i·7-s − 1.83·8-s + (−2.96 + 0.465i)9-s + 1.71i·10-s + (−1.62 − 2.89i)11-s + (−0.124 − 1.59i)12-s − 2.91i·13-s + 1.71i·14-s + (1.72 − 0.134i)15-s − 4.99·16-s − 2.59·17-s + ⋯
L(s)  = 1  + 1.20·2-s + (−0.0778 − 0.996i)3-s + 0.462·4-s + 0.447i·5-s + (−0.0941 − 1.20i)6-s + 0.377i·7-s − 0.649·8-s + (−0.987 + 0.155i)9-s + 0.540i·10-s + (−0.490 − 0.871i)11-s + (−0.0360 − 0.461i)12-s − 0.807i·13-s + 0.457i·14-s + (0.445 − 0.0348i)15-s − 1.24·16-s − 0.629·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.830 + 0.556i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.830 + 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.569159584\)
\(L(\frac12)\) \(\approx\) \(1.569159584\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.134 + 1.72i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
11 \( 1 + (1.62 + 2.89i)T \)
good2 \( 1 - 1.71T + 2T^{2} \)
13 \( 1 + 2.91iT - 13T^{2} \)
17 \( 1 + 2.59T + 17T^{2} \)
19 \( 1 + 8.54iT - 19T^{2} \)
23 \( 1 + 5.47iT - 23T^{2} \)
29 \( 1 + 3.64T + 29T^{2} \)
31 \( 1 + 0.818T + 31T^{2} \)
37 \( 1 - 7.87T + 37T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 + 7.69iT - 43T^{2} \)
47 \( 1 - 8.33iT - 47T^{2} \)
53 \( 1 - 8.95iT - 53T^{2} \)
59 \( 1 - 6.09iT - 59T^{2} \)
61 \( 1 - 8.94iT - 61T^{2} \)
67 \( 1 - 6.60T + 67T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 - 5.85iT - 73T^{2} \)
79 \( 1 - 5.16iT - 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + 1.96iT - 89T^{2} \)
97 \( 1 - 1.84T + 97T^{2} \)
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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

−9.197556646598071153745626298026, −8.606907894723306727937505125757, −7.58997062797574195907837173435, −6.69867911775133270365370245239, −5.97419286803920342555458347448, −5.34700906845793938277862132396, −4.29443699099220470467171563620, −2.78359755947940512635745225417, −2.67384036961628155288988160464, −0.43011696287748261867227939126, 2.08216812085565760519465701074, 3.52721214539195369340774063042, 4.10696382905965712806312321664, 4.83693581706140329709141013373, 5.58148838714918212677983076291, 6.38805014918353434804642978953, 7.62587522276955040301336146469, 8.588686067996105918153590179584, 9.628245077479366814787758061430, 9.886082746814760454140046765053

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