↑

Properties

Label 2-876-219.134-c0-0-0
Degree $2$
Conductor $876$
Sign $0.705 + 0.709i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)3-s + (0.296 − 1.10i)7-s + (0.499 − 0.866i)9-s + (−0.123 + 1.40i)13-s + (−1.85 + 0.326i)19-s + (−0.296 − 1.10i)21-s + (0.342 − 0.939i)25-s − 0.999i·27-s + (0.157 − 0.0736i)31-s + (−0.300 + 1.70i)37-s + (0.597 + 1.28i)39-s + (1.75 + 0.469i)43-s + (−0.273 − 0.157i)49-s + (−1.43 + 1.20i)57-s + (−0.826 + 0.984i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.296 − 1.10i)7-s + (0.499 − 0.866i)9-s + (−0.123 + 1.40i)13-s + (−1.85 + 0.326i)19-s + (−0.296 − 1.10i)21-s + (0.342 − 0.939i)25-s − 0.999i·27-s + (0.157 − 0.0736i)31-s + (−0.300 + 1.70i)37-s + (0.597 + 1.28i)39-s + (1.75 + 0.469i)43-s + (−0.273 − 0.157i)49-s + (−1.43 + 1.20i)57-s + (−0.826 + 0.984i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 876 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(876\)    =    \(2^{2} \cdot 3 \cdot 73\)
Sign: $0.705 + 0.709i$
Analytic conductor: \(0.437180\)
Root analytic conductor: \(0.661196\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{876} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 876,\ (\ :0),\ 0.705 + 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.308613131\)
\(L(\frac12)\) \(\approx\) \(1.308613131\)
\(L(1)\) 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 \( 1 + (-0.342 + 0.939i)T^{2} \)
7 \( 1 + (-0.296 + 1.10i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (0.342 - 0.939i)T^{2} \)
13 \( 1 + (0.123 - 1.40i)T + (-0.984 - 0.173i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (1.85 - 0.326i)T + (0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.342 - 0.939i)T^{2} \)
31 \( 1 + (-0.157 + 0.0736i)T + (0.642 - 0.766i)T^{2} \)
37 \( 1 + (0.300 - 1.70i)T + (-0.939 - 0.342i)T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-1.75 - 0.469i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.984 + 0.173i)T^{2} \)
53 \( 1 + (0.342 + 0.939i)T^{2} \)
59 \( 1 + (-0.984 - 0.173i)T^{2} \)
61 \( 1 + (0.826 - 0.984i)T + (-0.173 - 0.984i)T^{2} \)
67 \( 1 + (1.26 + 1.50i)T + (-0.173 + 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
79 \( 1 + (-0.984 - 1.17i)T + (-0.173 + 0.984i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.173 + 0.984i)T^{2} \)
97 \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.22656247614844636599649143321, −9.280271741214677587915211921509, −8.491094668890295284580541647879, −7.76870319138601822882585656689, −6.83385111881691503324264037052, −6.32356256107942262188436264204, −4.45115066688578724488460004612, −4.06829214378873057629845337634, −2.60490779707838394867947567037, −1.47154917226766950370732396638, 2.10229552951848598092955645971, 2.92696092277547738590267655081, 4.09290223032733656769436419117, 5.15317159642019660541715900123, 5.92552687543791116258180212423, 7.29165505787758912864331285686, 8.130009926809492125635931005546, 8.836166026572757385339557170866, 9.365933305667532738476459986710, 10.61608577322415500579278007059

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