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Properties

Label 2-1776-37.10-c1-0-9
Degree $2$
Conductor $1776$
Sign $0.982 + 0.188i$
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 + (0.169 + 0.293i)5-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s − 4.56·11-s + (0.669 + 1.15i)13-s + (0.169 − 0.293i)15-s + (−0.450 + 0.780i)17-s + (−2.62 − 4.53i)19-s + (0.499 − 0.866i)21-s + 8.14·23-s + (2.44 − 4.23i)25-s + 0.999·27-s + 10.2·29-s − 3.90·31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.0757 + 0.131i)5-s + (0.188 + 0.327i)7-s + (−0.166 + 0.288i)9-s − 1.37·11-s + (0.185 + 0.321i)13-s + (0.0437 − 0.0757i)15-s + (−0.109 + 0.189i)17-s + (−0.601 − 1.04i)19-s + (0.109 − 0.188i)21-s + 1.69·23-s + (0.488 − 0.846i)25-s + 0.192·27-s + 1.89·29-s − 0.700·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.982 + 0.188i$
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: \((2,\ 1776,\ (\ :1/2),\ 0.982 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.450182759\)
\(L(\frac12)\) \(\approx\) \(1.450182759\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 - 5.97i)T \)
good5 \( 1 + (-0.169 - 0.293i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.56T + 11T^{2} \)
13 \( 1 + (-0.669 - 1.15i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.450 - 0.780i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.62 + 4.53i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.14T + 23T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 + 3.90T + 31T^{2} \)
41 \( 1 + (-6.18 - 10.7i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 0.661T + 43T^{2} \)
47 \( 1 - 8.22T + 47T^{2} \)
53 \( 1 + (3.78 - 6.56i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.94 + 3.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.56 + 9.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.44 - 5.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.32 + 7.48i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.0T + 73T^{2} \)
79 \( 1 + (-5.02 - 8.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.57 + 7.93i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.24 - 5.61i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.20T + 97T^{2} \)
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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

−9.102388104486426515516208218997, −8.435524388284128114660869597184, −7.71137412965316314142688480047, −6.76745142169058803483842107626, −6.23292818408329910551528996803, −5.05896699055099946435065437492, −4.64828383429221604117718306906, −2.97023159816689750424795857412, −2.36875637289652825519733193843, −0.854403311154069586779839606933, 0.819331120756592914599002375588, 2.42301479548695901289699559429, 3.42001155463384548013245313843, 4.47244434880058473777445004744, 5.24539342688820542947887785429, 5.86825509059318033884899395401, 7.03550133482824572868976369850, 7.70625379774252456119432581989, 8.641509805662095549867224936638, 9.245568083121530336204672536052

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