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Properties

Label 2-1155-385.384-c1-0-1
Degree $2$
Conductor $1155$
Sign $-0.996 - 0.0845i$
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  + 0.226·2-s − 3-s − 1.94·4-s + (0.777 + 2.09i)5-s − 0.226·6-s + (−0.735 − 2.54i)7-s − 0.894·8-s + 9-s + (0.176 + 0.475i)10-s + (2.49 + 2.19i)11-s + 1.94·12-s + 3.41i·13-s + (−0.166 − 0.576i)14-s + (−0.777 − 2.09i)15-s + 3.69·16-s − 1.69i·17-s + ⋯
L(s)  = 1  + 0.160·2-s − 0.577·3-s − 0.974·4-s + (0.347 + 0.937i)5-s − 0.0925·6-s + (−0.277 − 0.960i)7-s − 0.316·8-s + 0.333·9-s + (0.0557 + 0.150i)10-s + (0.750 + 0.660i)11-s + 0.562·12-s + 0.948i·13-s + (−0.0445 − 0.153i)14-s + (−0.200 − 0.541i)15-s + 0.923·16-s − 0.410i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.996 - 0.0845i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.996 - 0.0845i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2577588158\)
\(L(\frac12)\) \(\approx\) \(0.2577588158\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + (-0.777 - 2.09i)T \)
7 \( 1 + (0.735 + 2.54i)T \)
11 \( 1 + (-2.49 - 2.19i)T \)
good2 \( 1 - 0.226T + 2T^{2} \)
13 \( 1 - 3.41iT - 13T^{2} \)
17 \( 1 + 1.69iT - 17T^{2} \)
19 \( 1 + 4.66T + 19T^{2} \)
23 \( 1 + 3.95iT - 23T^{2} \)
29 \( 1 + 3.66iT - 29T^{2} \)
31 \( 1 - 8.62iT - 31T^{2} \)
37 \( 1 + 3.09iT - 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 + 0.542T + 43T^{2} \)
47 \( 1 + 8.26T + 47T^{2} \)
53 \( 1 - 6.65iT - 53T^{2} \)
59 \( 1 - 5.11iT - 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 8.03iT - 67T^{2} \)
71 \( 1 + 3.92T + 71T^{2} \)
73 \( 1 + 0.227iT - 73T^{2} \)
79 \( 1 + 9.93iT - 79T^{2} \)
83 \( 1 + 14.3iT - 83T^{2} \)
89 \( 1 - 4.81iT - 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−10.25031371751959809783881543403, −9.524680870042074639010514854174, −8.741895702421547654744230004271, −7.48947195141769505398360985875, −6.67312162678459055622733427471, −6.24380052591179995615672364494, −4.82735713679786691156315770897, −4.25987398379105621528762779544, −3.30006196365276674462929618830, −1.65770531811284565012676495171, 0.12118853132988218105470529524, 1.56872051788997449850283717971, 3.28904565447717236858856449772, 4.30064995050164679969118143768, 5.28509557038834115377407259005, 5.73911897083048191110124190059, 6.52980788216950818665908555879, 8.261060296971295788318214182554, 8.458795187919458356043588922459, 9.479412757507184438270732103957

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