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Properties

Label 2-1776-1776.1109-c0-0-10
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  + (0.555 + 0.831i)2-s + (0.707 − 0.707i)3-s + (−0.382 + 0.923i)4-s + (−1.17 − 1.17i)5-s + (0.980 + 0.195i)6-s − 0.765i·7-s + (−0.980 + 0.195i)8-s − 1.00i·9-s + (0.324 − 1.63i)10-s + (0.382 + 0.923i)12-s + (0.636 − 0.425i)14-s − 1.66·15-s + (−0.707 − 0.707i)16-s − 1.11·17-s + (0.831 − 0.555i)18-s + ⋯
L(s)  = 1  + (0.555 + 0.831i)2-s + (0.707 − 0.707i)3-s + (−0.382 + 0.923i)4-s + (−1.17 − 1.17i)5-s + (0.980 + 0.195i)6-s − 0.765i·7-s + (−0.980 + 0.195i)8-s − 1.00i·9-s + (0.324 − 1.63i)10-s + (0.382 + 0.923i)12-s + (0.636 − 0.425i)14-s − 1.66·15-s + (−0.707 − 0.707i)16-s − 1.11·17-s + (0.831 − 0.555i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.231862068\)
\(L(\frac12)\) \(\approx\) \(1.231862068\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.555 - 0.831i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (1.17 + 1.17i)T + iT^{2} \)
7 \( 1 + 0.765iT - T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + 1.11T + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + 0.390iT - T^{2} \)
29 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-1.38 - 1.38i)T + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.84iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.96iT - T^{2} \)
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

−8.748379586947660261979401078785, −8.450301635400583308896772449835, −7.73649993762121224582681909823, −7.08472975156067819571857613847, −6.37571964539798059007648693278, −5.13797031182704928876068978038, −4.16381186346647939294739611149, −3.89556401952756321559801894798, −2.52639260924591240501805765923, −0.70814274130317515262556168233, 2.13767431588354542408431311146, 2.98425144473179747066025276064, 3.54271801144112640741976141410, 4.43935293678530130853760157390, 5.20206959581668352606606842164, 6.44213185005991434936674685452, 7.20853570150417628455912977403, 8.412255747135769962074903772506, 8.802814341631438434128702768209, 9.847732460999927010697124113588

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