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Properties

Label 2-1776-1776.1109-c0-0-5
Degree $2$
Conductor $1776$
Sign $-0.382 - 0.923i$
Analytic cond. $0.886339$
Root an. cond. $0.941456$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.195 + 0.980i)2-s + (−0.707 + 0.707i)3-s + (−0.923 − 0.382i)4-s + (1.38 + 1.38i)5-s + (−0.555 − 0.831i)6-s − 1.84i·7-s + (0.555 − 0.831i)8-s − 1.00i·9-s + (−1.63 + 1.08i)10-s + (0.923 − 0.382i)12-s + (1.81 + 0.360i)14-s − 1.96·15-s + (0.707 + 0.707i)16-s + 0.390·17-s + (0.980 + 0.195i)18-s + ⋯
L(s)  = 1  + (−0.195 + 0.980i)2-s + (−0.707 + 0.707i)3-s + (−0.923 − 0.382i)4-s + (1.38 + 1.38i)5-s + (−0.555 − 0.831i)6-s − 1.84i·7-s + (0.555 − 0.831i)8-s − 1.00i·9-s + (−1.63 + 1.08i)10-s + (0.923 − 0.382i)12-s + (1.81 + 0.360i)14-s − 1.96·15-s + (0.707 + 0.707i)16-s + 0.390·17-s + (0.980 + 0.195i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ -0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9638864203\)
\(L(\frac12)\) \(\approx\) \(0.9638864203\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.195 - 0.980i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
good5 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
7 \( 1 + 1.84iT - T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - 0.390T + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - 1.66iT - T^{2} \)
29 \( 1 + (-0.785 + 0.785i)T - iT^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-0.785 - 0.785i)T + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 0.765iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 1.11iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.911845646713584338977228567480, −9.288643118301453701026757832653, −7.80931379471651596765559833996, −7.12719939190231346373171075367, −6.52418976884809339273440601467, −5.91017091892861902616565598255, −5.07522042643046352001025955708, −4.04568822780110197380808186275, −3.24802657456442664786871954002, −1.28030760729925545500521746987, 1.05970025246878750076899446581, 2.11578287468415521169131486759, 2.59050433472454220408854047316, 4.58264770574559277523129053304, 5.24375410306373901910218154240, 5.72657381479531624673703925482, 6.54110866964944556101473462862, 8.223654771345751650830550500502, 8.543570148611992013974108512749, 9.270399628669420962617443969597

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