Properties

Label 2-1155-77.76-c1-0-63
Degree $2$
Conductor $1155$
Sign $0.727 - 0.686i$
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  − 2.23i·2-s i·3-s − 3.01·4-s i·5-s − 2.23·6-s + (−0.322 + 2.62i)7-s + 2.26i·8-s − 9-s − 2.23·10-s + (−2.55 − 2.11i)11-s + 3.01i·12-s − 0.842·13-s + (5.87 + 0.720i)14-s − 15-s − 0.950·16-s + 3.42·17-s + ⋯
L(s)  = 1  − 1.58i·2-s − 0.577i·3-s − 1.50·4-s − 0.447i·5-s − 0.914·6-s + (−0.121 + 0.992i)7-s + 0.801i·8-s − 0.333·9-s − 0.707·10-s + (−0.769 − 0.638i)11-s + 0.869i·12-s − 0.233·13-s + (1.57 + 0.192i)14-s − 0.258·15-s − 0.237·16-s + 0.829·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.727 - 0.686i$
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: \((2,\ 1155,\ (\ :1/2),\ 0.727 - 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1430336024\)
\(L(\frac12)\) \(\approx\) \(0.1430336024\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.322 - 2.62i)T \)
11 \( 1 + (2.55 + 2.11i)T \)
good2 \( 1 + 2.23iT - 2T^{2} \)
13 \( 1 + 0.842T + 13T^{2} \)
17 \( 1 - 3.42T + 17T^{2} \)
19 \( 1 + 6.30T + 19T^{2} \)
23 \( 1 + 5.97T + 23T^{2} \)
29 \( 1 - 9.40iT - 29T^{2} \)
31 \( 1 + 0.997iT - 31T^{2} \)
37 \( 1 - 1.41T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 + 8.10iT - 47T^{2} \)
53 \( 1 + 3.80T + 53T^{2} \)
59 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 - 3.15T + 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 - 0.946T + 83T^{2} \)
89 \( 1 - 17.4iT - 89T^{2} \)
97 \( 1 + 17.6iT - 97T^{2} \)
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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

−9.160769445326011643678100295454, −8.528431214447169437677653798321, −7.76275554825956830584531696418, −6.33973437195945318708530821286, −5.52623518272503608687857277874, −4.51419681786758934035441865338, −3.29372133264977083749579869767, −2.48582511239743017473250189031, −1.56314331666601529914172332697, −0.05946567660818507531167993059, 2.47259258670289368690266538255, 4.07562389191075230914021057266, 4.50030121573110548926121614283, 5.74167055892054152120389877504, 6.27515273698921652696440972960, 7.41614672084604332914128673284, 7.69458080332117538610862036173, 8.604518936349308481660894578859, 9.720957309615354971992962779198, 10.22719086483367820268180417525

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