Properties

Label 2-876-219.92-c0-0-0
Degree $2$
Conductor $876$
Sign $0.642 - 0.766i$
Analytic cond. $0.437180$
Root an. cond. $0.661196$
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  + (0.866 + 0.5i)3-s + (0.218 + 0.816i)7-s + (0.499 + 0.866i)9-s + (−1.15 + 0.811i)13-s + (−0.524 − 1.43i)19-s + (−0.218 + 0.816i)21-s + (0.642 − 0.766i)25-s + 0.999i·27-s + (−0.142 − 1.63i)31-s + (1.62 + 0.592i)37-s + (−1.40 + 0.123i)39-s + (−0.168 + 0.0451i)43-s + (0.247 − 0.142i)49-s + (0.266 − 1.50i)57-s + (−1.93 + 0.342i)61-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.218 + 0.816i)7-s + (0.499 + 0.866i)9-s + (−1.15 + 0.811i)13-s + (−0.524 − 1.43i)19-s + (−0.218 + 0.816i)21-s + (0.642 − 0.766i)25-s + 0.999i·27-s + (−0.142 − 1.63i)31-s + (1.62 + 0.592i)37-s + (−1.40 + 0.123i)39-s + (−0.168 + 0.0451i)43-s + (0.247 − 0.142i)49-s + (0.266 − 1.50i)57-s + (−1.93 + 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(876\)    =    \(2^{2} \cdot 3 \cdot 73\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(0.437180\)
Root analytic conductor: \(0.661196\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{876} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 876,\ (\ :0),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.284052998\)
\(L(\frac12)\) \(\approx\) \(1.284052998\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + iT \)
good5 \( 1 + (-0.642 + 0.766i)T^{2} \)
7 \( 1 + (-0.218 - 0.816i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.642 - 0.766i)T^{2} \)
13 \( 1 + (1.15 - 0.811i)T + (0.342 - 0.939i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.524 + 1.43i)T + (-0.766 + 0.642i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.642 - 0.766i)T^{2} \)
31 \( 1 + (0.142 + 1.63i)T + (-0.984 + 0.173i)T^{2} \)
37 \( 1 + (-1.62 - 0.592i)T + (0.766 + 0.642i)T^{2} \)
41 \( 1 + (0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.168 - 0.0451i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.342 + 0.939i)T^{2} \)
53 \( 1 + (0.642 + 0.766i)T^{2} \)
59 \( 1 + (0.342 - 0.939i)T^{2} \)
61 \( 1 + (1.93 - 0.342i)T + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (0.673 + 0.118i)T + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (0.342 + 0.0603i)T + (0.939 + 0.342i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (-1.70 + 0.984i)T + (0.5 - 0.866i)T^{2} \)
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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

−10.30307716254044580651617073400, −9.369251982166857635972015424115, −9.017345019454203025108806052175, −8.046940666779029999418297720791, −7.23382900168659013278366173587, −6.15439318781586863626831710678, −4.83633821205297865438840438200, −4.36153357765731827025514669982, −2.81057924664315450848766762383, −2.19364054274912134156601082890, 1.39011864945304327806498006748, 2.73592855306068369472590667778, 3.72334611509445472983828971752, 4.77414850729590462360641319655, 6.01195323799810015195109739956, 7.14160735526857161757082313275, 7.64160108998842040475694310020, 8.405191715233195137108321915169, 9.381379160494905756985466830569, 10.18903602611792987853373704448

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