↑

Properties

Label 2-495-55.54-c2-0-3
Degree $2$
Conductor $495$
Sign $-0.999 + 0.0393i$
Analytic cond. $13.4877$
Root an. cond. $3.67257$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.23·2-s + 1.00·4-s + (2 + 4.58i)5-s − 11.1·7-s − 6.70·8-s + (4.47 + 10.2i)10-s + (4 − 10.2i)11-s − 8.94·13-s − 25.0·14-s − 19·16-s − 15.6·17-s + 10.2i·19-s + (2.00 + 4.58i)20-s + (8.94 − 22.9i)22-s − 27.4i·23-s + ⋯
L(s)  = 1  + 1.11·2-s + 0.250·4-s + (0.400 + 0.916i)5-s − 1.59·7-s − 0.838·8-s + (0.447 + 1.02i)10-s + (0.363 − 0.931i)11-s − 0.688·13-s − 1.78·14-s − 1.18·16-s − 0.920·17-s + 0.539i·19-s + (0.100 + 0.229i)20-s + (0.406 − 1.04i)22-s − 1.19i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0393i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(495\)    =    \(3^{2} \cdot 5 \cdot 11\)
Sign: $-0.999 + 0.0393i$
Analytic conductor: \(13.4877\)
Root analytic conductor: \(3.67257\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{495} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 495,\ (\ :1),\ -0.999 + 0.0393i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3156562581\)
\(L(\frac12)\) \(\approx\) \(0.3156562581\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2 - 4.58i)T \)
11 \( 1 + (-4 + 10.2i)T \)
good2 \( 1 - 2.23T + 4T^{2} \)
7 \( 1 + 11.1T + 49T^{2} \)
13 \( 1 + 8.94T + 169T^{2} \)
17 \( 1 + 15.6T + 289T^{2} \)
19 \( 1 - 10.2iT - 361T^{2} \)
23 \( 1 + 27.4iT - 529T^{2} \)
29 \( 1 - 10.2iT - 841T^{2} \)
31 \( 1 + 3T + 961T^{2} \)
37 \( 1 - 4.58iT - 1.36e3T^{2} \)
41 \( 1 - 20.4iT - 1.68e3T^{2} \)
43 \( 1 + 22.3T + 1.84e3T^{2} \)
47 \( 1 - 64.1iT - 2.20e3T^{2} \)
53 \( 1 + 4.58iT - 2.80e3T^{2} \)
59 \( 1 - 18T + 3.48e3T^{2} \)
61 \( 1 - 71.7iT - 3.72e3T^{2} \)
67 \( 1 - 27.4iT - 4.48e3T^{2} \)
71 \( 1 + 27T + 5.04e3T^{2} \)
73 \( 1 + 58.1T + 5.32e3T^{2} \)
79 \( 1 + 61.4iT - 6.24e3T^{2} \)
83 \( 1 + 71.5T + 6.88e3T^{2} \)
89 \( 1 + 37T + 7.92e3T^{2} \)
97 \( 1 - 119. iT - 9.40e3T^{2} \)
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

−11.32303423503577293981850725832, −10.32087279168000475341521183627, −9.529699156918934882494503493907, −8.675301252705772907589129114999, −7.02573382524594838598421450333, −6.34136429822479655359214554957, −5.78404156615581566052892413825, −4.35010892858655956889383423495, −3.29226995427121400292943350867, −2.65685572249216599889185150263, 0.079968372685928295320777055885, 2.27365536415923108108243432451, 3.57178762590321475258741924921, 4.52633654737807756156526948189, 5.39607075478935583626232305218, 6.35947222441135673097491735291, 7.15675427137983809821372022363, 8.800951480946571216779074038387, 9.464201952526860652728242139876, 9.998152419811394938667759793491

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