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Properties

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

\(L(\frac{1}{2})\) \(\approx\) \(0.4263097983\)
\(L(\frac12)\) \(\approx\) \(0.4263097983\)
\(L(1)\) 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

−9.658747987184362745677861133103, −8.551608615424734559736826216685, −7.85547274390574139862574228903, −7.51664233918224992207891770982, −5.75867599637708723559704482841, −5.27814294075312994871661806594, −4.64340859393863741383835258909, −3.39134033125314452330725671541, −2.28890686749485397496995102585, −1.13028497845776133670328455561, 0.42647550358996798344830702770, 3.28670081519178969311908013997, 3.72361442772309986566122899159, 4.81824372620555751686067882807, 5.34135282259720531772792798359, 6.74449451010336901407839687137, 6.87609167957221890306513849067, 7.75318589797782235363108326631, 8.567062692902758512100159990008, 9.602275333745668947045904010406

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