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Properties

Label 2-876-219.23-c0-0-0
Degree $2$
Conductor $876$
Sign $0.353 + 0.935i$
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 + (−1.75 − 0.469i)7-s + (0.499 − 0.866i)9-s + (0.811 − 1.15i)13-s + (0.524 − 1.43i)19-s + (1.75 − 0.469i)21-s + (−0.642 − 0.766i)25-s + 0.999i·27-s + (1.14 + 0.0999i)31-s + (−1.62 + 0.592i)37-s + (−0.123 + 1.40i)39-s + (0.515 − 1.92i)43-s + (1.97 + 1.14i)49-s + (0.266 + 1.50i)57-s + (−1.93 − 0.342i)61-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−1.75 − 0.469i)7-s + (0.499 − 0.866i)9-s + (0.811 − 1.15i)13-s + (0.524 − 1.43i)19-s + (1.75 − 0.469i)21-s + (−0.642 − 0.766i)25-s + 0.999i·27-s + (1.14 + 0.0999i)31-s + (−1.62 + 0.592i)37-s + (−0.123 + 1.40i)39-s + (0.515 − 1.92i)43-s + (1.97 + 1.14i)49-s + (0.266 + 1.50i)57-s + (−1.93 − 0.342i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(876\)    =    \(2^{2} \cdot 3 \cdot 73\)
Sign: $0.353 + 0.935i$
Analytic conductor: \(0.437180\)
Root analytic conductor: \(0.661196\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{876} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 876,\ (\ :0),\ 0.353 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5223350999\)
\(L(\frac12)\) \(\approx\) \(0.5223350999\)
\(L(1)\) 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 \( 1 + (1.75 + 0.469i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.642 - 0.766i)T^{2} \)
13 \( 1 + (-0.811 + 1.15i)T + (-0.342 - 0.939i)T^{2} \)
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 \( 1 + (-1.14 - 0.0999i)T + (0.984 + 0.173i)T^{2} \)
37 \( 1 + (1.62 - 0.592i)T + (0.766 - 0.642i)T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.515 + 1.92i)T + (-0.866 - 0.5i)T^{2} \)
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

−10.40152779491713670441077037261, −9.530142949107183361492229433247, −8.762356065412307120247520617233, −7.38926680286785092736441039237, −6.54608773559188576540910849657, −5.96506240973009525407577417523, −4.93036266862055022660462011204, −3.74920111485222400758661093734, −3.02382324078786163815973953215, −0.59422924048075481769536626056, 1.62620754833653079943951388609, 3.15089782038192520479375227580, 4.20601262229024234232845153017, 5.64095011248212953410953765508, 6.18380626208322822779053201971, 6.83447699410312131738894333322, 7.81822427954232566411797427057, 8.998709564984188525008106641769, 9.733654061635297460820414831734, 10.46341229231642454633636056248

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