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Properties

Label 2-1155-77.76-c1-0-52
Degree $2$
Conductor $1155$
Sign $0.299 + 0.954i$
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  + 2.52i·2-s − i·3-s − 4.37·4-s + i·5-s + 2.52·6-s + (2.52 − 0.792i)7-s − 5.98i·8-s − 9-s − 2.52·10-s − 3.31i·11-s + 4.37i·12-s + (2 + 6.37i)14-s + 15-s + 6.37·16-s − 5.84·17-s − 2.52i·18-s + ⋯
L(s)  = 1  + 1.78i·2-s − 0.577i·3-s − 2.18·4-s + 0.447i·5-s + 1.03·6-s + (0.954 − 0.299i)7-s − 2.11i·8-s − 0.333·9-s − 0.798·10-s − 1.00i·11-s + 1.26i·12-s + (0.534 + 1.70i)14-s + 0.258·15-s + 1.59·16-s − 1.41·17-s − 0.594i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.299 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.299 + 0.954i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.299 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1021178314\)
\(L(\frac12)\) \(\approx\) \(0.1021178314\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 - iT \)
7 \( 1 + (-2.52 + 0.792i)T \)
11 \( 1 + 3.31iT \)
good2 \( 1 - 2.52iT - 2T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 5.84T + 17T^{2} \)
19 \( 1 + 7.72T + 19T^{2} \)
23 \( 1 + 8.11T + 23T^{2} \)
29 \( 1 - 2.67iT - 29T^{2} \)
31 \( 1 - 4.74iT - 31T^{2} \)
37 \( 1 + 0.744T + 37T^{2} \)
41 \( 1 + 8.21T + 41T^{2} \)
43 \( 1 + 5.84iT - 43T^{2} \)
47 \( 1 - 1.25iT - 47T^{2} \)
53 \( 1 + 7.37T + 53T^{2} \)
59 \( 1 - 7.37iT - 59T^{2} \)
61 \( 1 + 1.08T + 61T^{2} \)
67 \( 1 - 6.74T + 67T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 + 6.13T + 83T^{2} \)
89 \( 1 + 2.62iT - 89T^{2} \)
97 \( 1 - 2.62iT - 97T^{2} \)
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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

−9.058521015027676730996577161700, −8.339133884936218342142116157695, −8.070154970058242132934251421867, −6.92385027124466581678077439118, −6.51875554042896464836357263614, −5.70396085343173299569942572108, −4.71695867800393533133739643100, −3.83469231672564202586081188190, −2.03705716791270068385882166341, −0.04163142231475033150919254204, 1.90480416538569997001100902610, 2.29157558531542461076756333645, 3.98648125533046935450176735482, 4.39088167550012416187508910590, 5.10567280674492506773702203063, 6.43714901927980134522629198837, 8.142880952928763063850231189757, 8.524105476212985166445929417127, 9.532647385708698922202040147939, 9.967321992193620276811840444608

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