Properties

Label 2-1155-385.384-c1-0-78
Degree $2$
Conductor $1155$
Sign $-0.700 + 0.713i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.700 + 0.713i$
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: \((2,\ 1155,\ (\ :1/2),\ -0.700 + 0.713i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.059082493\)
\(L(\frac12)\) \(\approx\) \(1.059082493\)
\(L(\frac{3}{2})\) 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

−9.136188022961021890829683825229, −8.839523857048665361752804244095, −8.069517505644670696252284800534, −7.26192737615664884537786033468, −6.13685958217452126013237952869, −4.76681676656089289177828414113, −4.28026774158452474618815837889, −3.57452303962226877160719172405, −1.68689676661655259389353197366, −0.48348256008960172629515311757, 1.73918912466217670230544187627, 3.01437761254423579430679313971, 3.87028217019915803654862541336, 4.84251097681655085332672249490, 5.89641070808553534627792180053, 6.90248646911760264517914077327, 7.87945275529224919128706584381, 8.492265826161603332242576211329, 9.152696805802678884840232032669, 10.08764226304369259532289528625

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