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Properties

Label 2-1776-1776.1109-c0-0-1
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  + (−0.980 − 0.195i)2-s + (−0.707 + 0.707i)3-s + (0.923 + 0.382i)4-s + (−0.275 − 0.275i)5-s + (0.831 − 0.555i)6-s + 1.84i·7-s + (−0.831 − 0.555i)8-s − 1.00i·9-s + (0.216 + 0.324i)10-s + (−0.923 + 0.382i)12-s + (0.360 − 1.81i)14-s + 0.390·15-s + (0.707 + 0.707i)16-s + 1.96·17-s + (−0.195 + 0.980i)18-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)2-s + (−0.707 + 0.707i)3-s + (0.923 + 0.382i)4-s + (−0.275 − 0.275i)5-s + (0.831 − 0.555i)6-s + 1.84i·7-s + (−0.831 − 0.555i)8-s − 1.00i·9-s + (0.216 + 0.324i)10-s + (−0.923 + 0.382i)12-s + (0.360 − 1.81i)14-s + 0.390·15-s + (0.707 + 0.707i)16-s + 1.96·17-s + (−0.195 + 0.980i)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

\(L(\frac{1}{2})\) \(\approx\) \(0.4923307168\)
\(L(\frac12)\) \(\approx\) \(0.4923307168\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.980 + 0.195i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
good5 \( 1 + (0.275 + 0.275i)T + iT^{2} \)
7 \( 1 - 1.84iT - T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - 1.96T + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - 1.11iT - T^{2} \)
29 \( 1 + (1.17 - 1.17i)T - iT^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
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 \( 1 - 1.66iT - T^{2} \)
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

−9.571489419575200660588618546370, −9.178397423389535001201602135234, −8.334031343708276164069205819264, −7.62316765545468170895625608919, −6.41944293771692094205157021741, −5.67489486084000528241489521610, −5.15529655371131100408640685607, −3.64493549300489897673076192919, −2.86123078219300291175488829332, −1.44079833150842231491532780853, 0.61400591439394806388883945436, 1.60345555165618266592189155287, 3.11212006858778800265469518708, 4.26050939307056139628387997807, 5.51257893240109352991980031798, 6.28500984738639207262158636247, 7.15175439465496893152186932366, 7.64618858013363218870505289698, 7.964906498272634811385811120876, 9.357010492956383983229421212020

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