Properties

Label 2-1776-1776.1109-c0-0-6
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

\(L(\frac{1}{2})\) \(\approx\) \(1.509207006\)
\(L(\frac12)\) \(\approx\) \(1.509207006\)
\(L(1)\) 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

−9.698890298072954460877870737396, −8.958435820172427331157206344924, −8.393014511717577178454388738189, −7.47632081438894561562739199847, −6.51916864735104308332892407618, −5.84993479950917938516597283666, −5.20125335017672621399035988200, −4.07212914319151142534870031858, −3.03767223399095601635880928603, −2.23221680913406773634352888646, 1.20350202001712558603317857059, 1.80472073409812947468331512827, 2.87290515009225773391670960859, 4.15046870717702239151705133913, 4.67338162506538458044177612294, 5.80598979799368224361968317336, 7.00952058170054677866723705376, 7.52615655174249543087628468813, 8.856479840603154809450966740309, 9.176783102513466926566882858029

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