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Properties

Label 2-1155-77.76-c1-0-54
Degree $2$
Conductor $1155$
Sign $-0.990 + 0.134i$
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  − 1.26i·2-s − i·3-s + 0.399·4-s − i·5-s − 1.26·6-s + (0.0556 − 2.64i)7-s − 3.03i·8-s − 9-s − 1.26·10-s + (−0.516 − 3.27i)11-s − 0.399i·12-s + 6.21·13-s + (−3.34 − 0.0703i)14-s − 15-s − 3.04·16-s + 1.62·17-s + ⋯
L(s)  = 1  − 0.894i·2-s − 0.577i·3-s + 0.199·4-s − 0.447i·5-s − 0.516·6-s + (0.0210 − 0.999i)7-s − 1.07i·8-s − 0.333·9-s − 0.400·10-s + (−0.155 − 0.987i)11-s − 0.115i·12-s + 1.72·13-s + (−0.894 − 0.0188i)14-s − 0.258·15-s − 0.760·16-s + 0.394·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.990 + 0.134i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.990 + 0.134i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.978135689\)
\(L(\frac12)\) \(\approx\) \(1.978135689\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0556 + 2.64i)T \)
11 \( 1 + (0.516 + 3.27i)T \)
good2 \( 1 + 1.26iT - 2T^{2} \)
13 \( 1 - 6.21T + 13T^{2} \)
17 \( 1 - 1.62T + 17T^{2} \)
19 \( 1 - 4.38T + 19T^{2} \)
23 \( 1 + 1.24T + 23T^{2} \)
29 \( 1 - 9.91iT - 29T^{2} \)
31 \( 1 - 7.78iT - 31T^{2} \)
37 \( 1 - 1.61T + 37T^{2} \)
41 \( 1 + 4.95T + 41T^{2} \)
43 \( 1 + 3.35iT - 43T^{2} \)
47 \( 1 - 4.74iT - 47T^{2} \)
53 \( 1 + 8.51T + 53T^{2} \)
59 \( 1 + 4.49iT - 59T^{2} \)
61 \( 1 + 6.43T + 61T^{2} \)
67 \( 1 - 3.96T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 - 8.16T + 73T^{2} \)
79 \( 1 + 13.4iT - 79T^{2} \)
83 \( 1 + 2.78T + 83T^{2} \)
89 \( 1 + 2.92iT - 89T^{2} \)
97 \( 1 - 15.7iT - 97T^{2} \)
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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

−9.492525292141454229855174823039, −8.605238346937424733787816424593, −7.81887294654828539325867900274, −6.87041670020596180908815304459, −6.16632620146086866831431312821, −5.05652318725731382548763252696, −3.54306296530200522639970990502, −3.29642223735779898227687319948, −1.50161432441550814952964757717, −0.950689581700310294889015107429, 1.98264035501679029567070951613, 3.04294080254156847918050351743, 4.23587276075317620025147893061, 5.43010577178740804187007819301, 5.96471331786567530304038788831, 6.70775088805493822538509866888, 7.86355153057646248437770429891, 8.259909346743313108373370789320, 9.392085531539270657163231496295, 9.959755119138677948614631340063

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