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Properties

Label 2-1776-444.311-c0-0-1
Degree $2$
Conductor $1776$
Sign $0.952 + 0.303i$
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  + (0.939 − 0.342i)3-s + (1.50 + 0.266i)7-s + (0.766 − 0.642i)9-s + (−0.173 + 0.0151i)13-s + (−1.75 − 0.816i)19-s + (1.50 − 0.266i)21-s + (−0.342 + 0.939i)25-s + (0.500 − 0.866i)27-s + (−1.15 + 1.15i)31-s + (0.5 + 0.866i)37-s + (−0.157 + 0.0736i)39-s + (−1.28 − 1.28i)43-s + (1.26 + 0.460i)49-s + (−1.92 − 0.168i)57-s + (0.123 + 1.40i)61-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)3-s + (1.50 + 0.266i)7-s + (0.766 − 0.642i)9-s + (−0.173 + 0.0151i)13-s + (−1.75 − 0.816i)19-s + (1.50 − 0.266i)21-s + (−0.342 + 0.939i)25-s + (0.500 − 0.866i)27-s + (−1.15 + 1.15i)31-s + (0.5 + 0.866i)37-s + (−0.157 + 0.0736i)39-s + (−1.28 − 1.28i)43-s + (1.26 + 0.460i)49-s + (−1.92 − 0.168i)57-s + (0.123 + 1.40i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.952 + 0.303i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (1199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ 0.952 + 0.303i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.756286959\)
\(L(\frac12)\) \(\approx\) \(1.756286959\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.342 - 0.939i)T^{2} \)
7 \( 1 + (-1.50 - 0.266i)T + (0.939 + 0.342i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.173 - 0.0151i)T + (0.984 - 0.173i)T^{2} \)
17 \( 1 + (0.984 + 0.173i)T^{2} \)
19 \( 1 + (1.75 + 0.816i)T + (0.642 + 0.766i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T^{2} \)
31 \( 1 + (1.15 - 1.15i)T - iT^{2} \)
41 \( 1 + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (1.28 + 1.28i)T + iT^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.939 - 0.342i)T^{2} \)
59 \( 1 + (0.342 + 0.939i)T^{2} \)
61 \( 1 + (-0.123 - 1.40i)T + (-0.984 + 0.173i)T^{2} \)
67 \( 1 + (-0.118 + 0.673i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.766 - 0.642i)T^{2} \)
73 \( 1 + 1.87iT - T^{2} \)
79 \( 1 + (-0.692 + 0.484i)T + (0.342 - 0.939i)T^{2} \)
83 \( 1 + (0.173 - 0.984i)T^{2} \)
89 \( 1 + (0.342 + 0.939i)T^{2} \)
97 \( 1 + (-0.218 - 0.816i)T + (-0.866 + 0.5i)T^{2} \)
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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

−9.023094480609193865542538879580, −8.728840355905285417983694240454, −7.947286509931876757550686239212, −7.26086079244691531180031180871, −6.43490504412050052364748091748, −5.19547471259148090993046524195, −4.50138986951046833944001575326, −3.47886702337314577913101716939, −2.26907342646197950464831062592, −1.59003897160866546343857871910, 1.71320374511198935701899352593, 2.40205376479599517580373046909, 3.87783608174310096693718899232, 4.35728803097558995360961458348, 5.25646123915542125993330098737, 6.37933151782182584360740422583, 7.48716826469850414876761933460, 8.113989601731059614103087464584, 8.494845633820649619599326882305, 9.485226372090951962097147890465

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