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Properties

Label 2-1776-37.10-c1-0-30
Degree $2$
Conductor $1776$
Sign $-0.887 + 0.461i$
Analytic cond. $14.1814$
Root an. cond. $3.76582$
Motivic weight $1$
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 + (1.62 + 2.81i)5-s + (−1.5 − 2.59i)7-s + (−0.499 + 0.866i)9-s − 4.55·11-s + (0.124 + 0.215i)13-s + (1.62 − 2.81i)15-s + (2.90 − 5.02i)17-s + (1.27 + 2.21i)19-s + (−1.5 + 2.59i)21-s + 3.24·23-s + (−2.77 + 4.81i)25-s + 0.999·27-s − 10.3·29-s − 6.80·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.726 + 1.25i)5-s + (−0.566 − 0.981i)7-s + (−0.166 + 0.288i)9-s − 1.37·11-s + (0.0345 + 0.0597i)13-s + (0.419 − 0.726i)15-s + (0.703 − 1.21i)17-s + (0.293 + 0.507i)19-s + (−0.327 + 0.566i)21-s + 0.677·23-s + (−0.555 + 0.962i)25-s + 0.192·27-s − 1.91·29-s − 1.22·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.887 + 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $-0.887 + 0.461i$
Analytic conductor: \(14.1814\)
Root analytic conductor: \(3.76582\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :1/2),\ -0.887 + 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4560399591\)
\(L(\frac12)\) \(\approx\) \(0.4560399591\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.59 + 3.98i)T \)
good5 \( 1 + (-1.62 - 2.81i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.55T + 11T^{2} \)
13 \( 1 + (-0.124 - 0.215i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.90 + 5.02i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.27 - 2.21i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.24T + 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 + 6.80T + 31T^{2} \)
41 \( 1 + (-0.277 - 0.480i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + (-0.653 + 1.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.27 - 2.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.55 + 9.62i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.02 - 1.77i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.49 + 12.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 3.30T + 73T^{2} \)
79 \( 1 + (-2.02 - 3.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4 + 6.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.24 + 7.35i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.05T + 97T^{2} \)
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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

−9.174737105065287899356674835367, −7.66511522335627598364330127058, −7.37040087727222475902497621262, −6.73849851434085086689048999413, −5.72714071266517177942141087260, −5.18646028458631841640711295567, −3.59710310318644247578114859508, −2.92119575125639303646413484739, −1.85348599001604341133206537898, −0.16512710201240284530371306530, 1.55471640104107970981544096014, 2.72264104481749349554242835753, 3.81742915884570167866115583212, 5.13279943351710455220992697701, 5.42611697538908640007680168017, 5.99931026748684616326405217262, 7.30035310355909307985373244081, 8.329760768572756134735773754389, 8.975522220655598788199042433952, 9.491125414188737440926462001204

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