Properties

Label 2-1776-111.47-c0-0-0
Degree $2$
Conductor $1776$
Sign $0.367 - 0.929i$
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  + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)13-s + (1 + 1.73i)19-s + (0.499 − 0.866i)21-s + (−0.5 + 0.866i)25-s − 0.999·27-s + 31-s + 37-s + (−0.499 + 0.866i)39-s + 43-s + (−0.999 + 1.73i)57-s + (−1 − 1.73i)61-s + 0.999·63-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)13-s + (1 + 1.73i)19-s + (0.499 − 0.866i)21-s + (−0.5 + 0.866i)25-s − 0.999·27-s + 31-s + 37-s + (−0.499 + 0.866i)39-s + 43-s + (−0.999 + 1.73i)57-s + (−1 − 1.73i)61-s + 0.999·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.367 - 0.929i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ 0.367 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.267481890\)
\(L(\frac12)\) \(\approx\) \(1.267481890\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.677853962000689571077258512864, −9.042186843722134281448232793420, −8.010529295559822910899513070243, −7.49246560253283693970212461781, −6.36920046172565192977629468872, −5.56949854893600664994369288042, −4.42813594897073325934726593499, −3.80568205553149369716010253161, −3.05718088586235969492704925960, −1.59841125263865401988053918924, 0.994463184971430736708565272484, 2.64328669925049778356037693813, 2.89681049440655625679880817407, 4.28065774660506646052382418577, 5.57084126838916633563934531444, 6.12675091943907365783702913901, 7.01151212015942324711392525753, 7.77922638635996294022720573281, 8.559666746269037484949126565118, 9.170672477062316034205995503717

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