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Properties

Label 2-1155-77.76-c1-0-56
Degree $2$
Conductor $1155$
Sign $-0.360 - 0.932i$
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  − 1.93i·2-s − i·3-s − 1.73·4-s + i·5-s − 1.93·6-s + (0.189 − 2.63i)7-s − 0.517i·8-s − 9-s + 1.93·10-s + (−3 + 1.41i)11-s + 1.73i·12-s + 4.24·13-s + (−5.09 − 0.366i)14-s + 15-s − 4.46·16-s − 2.44·17-s + ⋯
L(s)  = 1  − 1.36i·2-s − 0.577i·3-s − 0.866·4-s + 0.447i·5-s − 0.788·6-s + (0.0716 − 0.997i)7-s − 0.183i·8-s − 0.333·9-s + 0.610·10-s + (−0.904 + 0.426i)11-s + 0.500i·12-s + 1.17·13-s + (−1.36 − 0.0978i)14-s + 0.258·15-s − 1.11·16-s − 0.594·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.360 - 0.932i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.360 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8860241434\)
\(L(\frac12)\) \(\approx\) \(0.8860241434\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.189 + 2.63i)T \)
11 \( 1 + (3 - 1.41i)T \)
good2 \( 1 + 1.93iT - 2T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 + 3.86T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 3.86iT - 29T^{2} \)
31 \( 1 + 10.9iT - 31T^{2} \)
37 \( 1 + 7.46T + 37T^{2} \)
41 \( 1 - 1.79T + 41T^{2} \)
43 \( 1 + 2.44iT - 43T^{2} \)
47 \( 1 - 10.9iT - 47T^{2} \)
53 \( 1 - 3.46T + 53T^{2} \)
59 \( 1 - 0.928iT - 59T^{2} \)
61 \( 1 - 4.89T + 61T^{2} \)
67 \( 1 - 1.46T + 67T^{2} \)
71 \( 1 + 6.92T + 71T^{2} \)
73 \( 1 - 9.14T + 73T^{2} \)
79 \( 1 - 1.03iT - 79T^{2} \)
83 \( 1 - 1.41T + 83T^{2} \)
89 \( 1 - 8.92iT - 89T^{2} \)
97 \( 1 + 8.92iT - 97T^{2} \)
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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

−9.590497096109653580395885508575, −8.433025816236083907669528233468, −7.64285690630611789504400070083, −6.75987996743773272726978263095, −5.94873908598800186761249312827, −4.38457350821413513539872786005, −3.77175127203351641141120407645, −2.57733452536802176144269941421, −1.76736893517828755608347368942, −0.36002954087178937653251475552, 2.12581333129732122319476277510, 3.50568180735114969678819225252, 4.78884557723422915740100226903, 5.39346776841431334299930445031, 6.09069656735545000702272546793, 6.88907419303781690478393536131, 8.170039440768901010966602996001, 8.613323739797205580497250853620, 8.973029117283842414608164010012, 10.32132773949081480205276228502

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