Properties

Label 2-1776-37.27-c1-0-22
Degree $2$
Conductor $1776$
Sign $-0.367 + 0.929i$
Analytic cond. $14.1814$
Root an. cond. $3.76582$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−3 + 1.73i)5-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s − 6·11-s + (1.5 − 0.866i)13-s + (−3 − 1.73i)15-s + (6 + 3.46i)17-s + (−3 + 1.73i)19-s + (−0.499 + 0.866i)21-s − 3.46i·23-s + (3.5 − 6.06i)25-s − 0.999·27-s + 6.92i·29-s + 1.73i·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−1.34 + 0.774i)5-s + (0.188 + 0.327i)7-s + (−0.166 + 0.288i)9-s − 1.80·11-s + (0.416 − 0.240i)13-s + (−0.774 − 0.447i)15-s + (1.45 + 0.840i)17-s + (−0.688 + 0.397i)19-s + (−0.109 + 0.188i)21-s − 0.722i·23-s + (0.700 − 1.21i)25-s − 0.192·27-s + 1.28i·29-s + 0.311i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $-0.367 + 0.929i$
Analytic conductor: \(14.1814\)
Root analytic conductor: \(3.76582\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1776,\ (\ :1/2),\ -0.367 + 0.929i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (5 + 3.46i)T \)
good5 \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6 - 3.46i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
41 \( 1 + (6 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 12.1iT - 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + (1.5 - 0.866i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-9 - 5.19i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.73iT - 97T^{2} \)
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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

−8.698040574203444117350388076744, −8.271886193440367420909765847762, −7.65576053407212149437387746036, −6.86180475129778439787213227881, −5.59800017376546958120928233100, −4.98473062494809732778200194214, −3.65907766483328818489774049568, −3.36291910475398319502504128260, −2.17272926869154637850649909301, 0, 1.20503928985695502022872761724, 2.74362771514290852962528570037, 3.59629402985586351891222112675, 4.68470205554538382148836130067, 5.25353694314861158507706981204, 6.47860027730196002756742771664, 7.56857163183667331135319514843, 7.995569572111011164870962261870, 8.284240739467849834831200801926

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