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Properties

Label 2-1776-1776.1109-c0-0-2
Degree $2$
Conductor $1776$
Sign $-0.923 - 0.382i$
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  + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (1 + i)5-s + (0.707 − 0.707i)6-s + i·7-s − i·8-s + 1.00i·9-s + (−1 + i)10-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s − 14-s − 1.41i·15-s + 16-s − 17-s + ⋯
L(s)  = 1  + i·2-s + (−0.707 − 0.707i)3-s − 4-s + (1 + i)5-s + (0.707 − 0.707i)6-s + i·7-s − i·8-s + 1.00i·9-s + (−1 + i)10-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s − 14-s − 1.41i·15-s + 16-s − 17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $-0.923 - 0.382i$
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: \((2,\ 1776,\ (\ :0),\ -0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7626651646\)
\(L(\frac12)\) \(\approx\) \(0.7626651646\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + (-1 - i)T + iT^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
13 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
59 \( 1 + (1 + i)T + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
89 \( 1 - iT - T^{2} \)
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

−9.675541007020531795517205010886, −8.891129755964836800497074914268, −8.168992447074487257522734041701, −7.05520497769877433643870628365, −6.53780422132570700655209188122, −6.16171698880504470814713027128, −5.22457236056839815773111680047, −4.51134668976022079174254044319, −2.73011817394263970643185529307, −1.90429934448867606057798773257, 0.62899797388846228784706406365, 1.80771923011138709800585607710, 3.32511347557105928233368470731, 4.12747979783663438427533168682, 4.96110290821543132743037624265, 5.63415295904561272792671217120, 6.32286749356017299576359837245, 7.916937878427065088608538987612, 8.653954892414451970993050078304, 9.392679238620304366614409345144

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