↑

Properties

Label 2-2e3-8.3-c18-0-10
Degree $2$
Conductor $8$
Sign $0.135 - 0.990i$
Analytic cond. $16.4308$
Root an. cond. $4.05350$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (431. + 275. i)2-s + 2.26e4·3-s + (1.10e5 + 2.37e5i)4-s + 1.13e5i·5-s + (9.76e6 + 6.24e6i)6-s + 3.55e7i·7-s + (−1.81e7 + 1.32e8i)8-s + 1.25e8·9-s + (−3.12e7 + 4.88e7i)10-s + 2.18e9·11-s + (2.49e9 + 5.38e9i)12-s − 7.03e9i·13-s + (−9.80e9 + 1.53e10i)14-s + 2.56e9i·15-s + (−4.45e10 + 5.23e10i)16-s + 6.55e10·17-s + ⋯
L(s)  = 1  + (0.842 + 0.538i)2-s + 1.15·3-s + (0.419 + 0.907i)4-s + 0.0579i·5-s + (0.969 + 0.619i)6-s + 0.880i·7-s + (−0.135 + 0.990i)8-s + 0.323·9-s + (−0.0312 + 0.0488i)10-s + 0.926·11-s + (0.482 + 1.04i)12-s − 0.663i·13-s + (−0.474 + 0.742i)14-s + 0.0666i·15-s + (−0.647 + 0.762i)16-s + 0.552·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.135 - 0.990i$
Analytic conductor: \(16.4308\)
Root analytic conductor: \(4.05350\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :9),\ 0.135 - 0.990i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(3.25469 + 2.84078i\)
\(L(\frac12)\) \(\approx\) \(3.25469 + 2.84078i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-431. - 275. i)T \)
good3 \( 1 - 2.26e4T + 3.87e8T^{2} \)
5 \( 1 - 1.13e5iT - 3.81e12T^{2} \)
7 \( 1 - 3.55e7iT - 1.62e15T^{2} \)
11 \( 1 - 2.18e9T + 5.55e18T^{2} \)
13 \( 1 + 7.03e9iT - 1.12e20T^{2} \)
17 \( 1 - 6.55e10T + 1.40e22T^{2} \)
19 \( 1 + 1.62e11T + 1.04e23T^{2} \)
23 \( 1 - 3.09e12iT - 3.24e24T^{2} \)
29 \( 1 + 2.64e13iT - 2.10e26T^{2} \)
31 \( 1 + 2.98e13iT - 6.99e26T^{2} \)
37 \( 1 + 1.12e14iT - 1.68e28T^{2} \)
41 \( 1 + 1.86e14T + 1.07e29T^{2} \)
43 \( 1 - 7.27e14T + 2.52e29T^{2} \)
47 \( 1 + 1.44e15iT - 1.25e30T^{2} \)
53 \( 1 + 6.79e14iT - 1.08e31T^{2} \)
59 \( 1 + 6.53e15T + 7.50e31T^{2} \)
61 \( 1 + 7.84e14iT - 1.36e32T^{2} \)
67 \( 1 - 1.65e16T + 7.40e32T^{2} \)
71 \( 1 - 1.64e16iT - 2.10e33T^{2} \)
73 \( 1 + 1.15e17T + 3.46e33T^{2} \)
79 \( 1 + 9.97e16iT - 1.43e34T^{2} \)
83 \( 1 - 2.58e17T + 3.49e34T^{2} \)
89 \( 1 - 1.02e17T + 1.22e35T^{2} \)
97 \( 1 + 4.73e17T + 5.77e35T^{2} \)
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

−17.24578345513114114548050307486, −15.43448473420006116760955418271, −14.62812821479187150320979750513, −13.34183059582920111494088366947, −11.80331732585820370790617081673, −9.091216679024333256519782629919, −7.76292428051014972840391989174, −5.80096098438818193978576835761, −3.70173887004536867009084682136, −2.36055373853317404353895792242, 1.32465865054460520225992010104, 3.03489798450891855575988342413, 4.36430974334292707882362270828, 6.80315596559768614937314241335, 8.952982447431650443872224649976, 10.65359332035581004118458063518, 12.50213700247764793781351136837, 14.08468562529720926858943875512, 14.57264276421209924975761905432, 16.55545477521051196365412753701

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