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Properties

Label 2-1776-444.143-c0-0-0
Degree $2$
Conductor $1776$
Sign $0.274 - 0.961i$
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 + 1.98i)13-s + (0.218 + 0.469i)19-s + (−1.50 − 0.266i)21-s + (0.342 + 0.939i)25-s + (0.500 + 0.866i)27-s + (0.811 − 0.811i)31-s + (0.5 − 0.866i)37-s + (−0.842 + 1.80i)39-s + (−0.597 − 0.597i)43-s + (1.26 − 0.460i)49-s + (0.0451 + 0.515i)57-s + (1.40 + 0.123i)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 + 1.98i)13-s + (0.218 + 0.469i)19-s + (−1.50 − 0.266i)21-s + (0.342 + 0.939i)25-s + (0.500 + 0.866i)27-s + (0.811 − 0.811i)31-s + (0.5 − 0.866i)37-s + (−0.842 + 1.80i)39-s + (−0.597 − 0.597i)43-s + (1.26 − 0.460i)49-s + (0.0451 + 0.515i)57-s + (1.40 + 0.123i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.274 - 0.961i$
Analytic conductor: \(0.886339\)
Root analytic conductor: \(0.941456\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :0),\ 0.274 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.293293974\)
\(L(\frac12)\) \(\approx\) \(1.293293974\)
\(L(1)\) 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 \( 1 + (0.173 - 1.98i)T + (-0.984 - 0.173i)T^{2} \)
17 \( 1 + (-0.984 + 0.173i)T^{2} \)
19 \( 1 + (-0.218 - 0.469i)T + (-0.642 + 0.766i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.866 + 0.5i)T^{2} \)
31 \( 1 + (-0.811 + 0.811i)T - iT^{2} \)
41 \( 1 + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (0.597 + 0.597i)T + iT^{2} \)
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 \( 1 + (-1.40 - 0.123i)T + (0.984 + 0.173i)T^{2} \)
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 \( 1 + (1.03 - 1.48i)T + (-0.342 - 0.939i)T^{2} \)
83 \( 1 + (0.173 + 0.984i)T^{2} \)
89 \( 1 + (-0.342 + 0.939i)T^{2} \)
97 \( 1 + (1.75 + 0.469i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.486137981686036091260211219222, −9.121245897832329818240227139447, −8.230347153909588303028351751076, −7.14372159765363088033873773840, −6.68559491838601463207942341673, −5.65640734853074596869178887674, −4.42290695012991334837207853534, −3.73754016333996615502520725813, −2.84769552003115266607703742867, −1.86480018275403179583629111225, 0.893521207859496099654916577543, 2.79805512921924789123143579391, 3.01114524461184346792667616847, 4.08379697550558220515377779668, 5.30096136627545925331100849675, 6.39344615365293745773858296989, 6.90070268963788034355375278755, 7.87514469130834849460976702986, 8.427896969521565442671144299952, 9.341727840723158817280468588327

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