↑

Properties

Label 2-837-93.47-c0-0-1
Degree $2$
Conductor $837$
Sign $0.817 + 0.575i$
Analytic cond. $0.417717$
Root an. cond. $0.646310$
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  + (0.363 + 0.5i)2-s + (0.190 − 0.587i)4-s − i·5-s + (0.951 − 0.309i)8-s + (0.5 − 0.363i)10-s + (−0.951 − 0.309i)11-s + (−1.53 + 0.5i)17-s + (0.809 − 0.587i)19-s + (−0.587 − 0.190i)20-s + (−0.190 − 0.587i)22-s + (0.587 − 0.190i)23-s + (0.587 + 0.809i)29-s + (−0.309 + 0.951i)31-s − i·32-s + (−0.809 − 0.587i)34-s + ⋯
L(s)  = 1  + (0.363 + 0.5i)2-s + (0.190 − 0.587i)4-s − i·5-s + (0.951 − 0.309i)8-s + (0.5 − 0.363i)10-s + (−0.951 − 0.309i)11-s + (−1.53 + 0.5i)17-s + (0.809 − 0.587i)19-s + (−0.587 − 0.190i)20-s + (−0.190 − 0.587i)22-s + (0.587 − 0.190i)23-s + (0.587 + 0.809i)29-s + (−0.309 + 0.951i)31-s − i·32-s + (−0.809 − 0.587i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.817 + 0.575i$
Analytic conductor: \(0.417717\)
Root analytic conductor: \(0.646310\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (512, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 837,\ (\ :0),\ 0.817 + 0.575i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.214526383\)
\(L(\frac12)\) \(\approx\) \(1.214526383\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + (0.309 - 0.951i)T \)
good2 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + iT - T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T^{2} \)
11 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
43 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
47 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + 0.618T + T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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

−10.53405557334770188766021132738, −9.288324027751002467936459272080, −8.719753498989186764201136434748, −7.66428319712651302144033304863, −6.78187315066921883365076731745, −5.85808944887798318124203274679, −4.95578351172148314550754836975, −4.49318171869524843977954770255, −2.76469392979477084172494406907, −1.23816960931090004644564952976, 2.25953810387835589274624185962, 2.87731325197656982465771155859, 3.98995982319281745693736392651, 4.95932514264699424641393511161, 6.21658146551958319795423029165, 7.27454903064555219833080469380, 7.65268038786427806397486293978, 8.817155030320425763916875421764, 9.886283575504817180925325244744, 10.80503068797354260084550986825

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