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Properties

Label 2-1155-33.32-c1-0-47
Degree $2$
Conductor $1155$
Sign $-0.0485 - 0.998i$
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  + 2.55·2-s + (−0.371 + 1.69i)3-s + 4.54·4-s + i·5-s + (−0.951 + 4.32i)6-s + i·7-s + 6.52·8-s + (−2.72 − 1.25i)9-s + 2.55i·10-s + (−3.20 + 0.868i)11-s + (−1.69 + 7.69i)12-s + 4.12i·13-s + 2.55i·14-s + (−1.69 − 0.371i)15-s + 7.59·16-s + 4.41·17-s + ⋯
L(s)  = 1  + 1.80·2-s + (−0.214 + 0.976i)3-s + 2.27·4-s + 0.447i·5-s + (−0.388 + 1.76i)6-s + 0.377i·7-s + 2.30·8-s + (−0.907 − 0.419i)9-s + 0.809i·10-s + (−0.965 + 0.261i)11-s + (−0.488 + 2.22i)12-s + 1.14i·13-s + 0.683i·14-s + (−0.436 − 0.0960i)15-s + 1.89·16-s + 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0485 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0485 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.0485 - 0.998i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.0485 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.246069675\)
\(L(\frac12)\) \(\approx\) \(4.246069675\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.371 - 1.69i)T \)
5 \( 1 - iT \)
7 \( 1 - iT \)
11 \( 1 + (3.20 - 0.868i)T \)
good2 \( 1 - 2.55T + 2T^{2} \)
13 \( 1 - 4.12iT - 13T^{2} \)
17 \( 1 - 4.41T + 17T^{2} \)
19 \( 1 + 1.90iT - 19T^{2} \)
23 \( 1 - 0.792iT - 23T^{2} \)
29 \( 1 + 2.29T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 - 3.49T + 37T^{2} \)
41 \( 1 + 0.636T + 41T^{2} \)
43 \( 1 + 9.10iT - 43T^{2} \)
47 \( 1 - 0.336iT - 47T^{2} \)
53 \( 1 + 11.0iT - 53T^{2} \)
59 \( 1 - 1.83iT - 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 - 9.49T + 67T^{2} \)
71 \( 1 + 5.28iT - 71T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 + 4.50iT - 79T^{2} \)
83 \( 1 + 2.03T + 83T^{2} \)
89 \( 1 - 6.21iT - 89T^{2} \)
97 \( 1 - 8.07T + 97T^{2} \)
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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

−10.28651579581256935602422081155, −9.478138264625930914125726471118, −8.224268301102042060251050457735, −7.11842096656416139015872582914, −6.28453661795623856139266318742, −5.48904564661811801024626909850, −4.84249953961052549175939692921, −4.01627028110279779359569102212, −3.10963705060629398048526671817, −2.29573077119670605594756428965, 1.08587173554979166230905407486, 2.57060299317882099781833520910, 3.26612192014130449143602078067, 4.55079647970935227368632225585, 5.43581343860232704162722228170, 5.88079546931216033133647783974, 6.80010502055671017653351335796, 7.905916799772035034941422031285, 8.056866418628188315435964060642, 9.921559471135118091371763816563

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