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Properties

Label 2-1155-33.32-c1-0-93
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

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

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