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Properties

Label 2-2e3-8.5-c21-0-9
Degree $2$
Conductor $8$
Sign $0.487 + 0.873i$
Analytic cond. $22.3581$
Root an. cond. $4.72844$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35e3 − 501. i)2-s + 1.31e5i·3-s + (1.59e6 + 1.36e6i)4-s − 4.61e6i·5-s + (6.59e7 − 1.78e8i)6-s − 9.67e8·7-s + (−1.48e9 − 2.65e9i)8-s − 6.79e9·9-s + (−2.31e9 + 6.26e9i)10-s + 2.63e10i·11-s + (−1.79e11 + 2.09e11i)12-s + 1.34e11i·13-s + (1.31e12 + 4.85e11i)14-s + 6.05e11·15-s + (6.81e11 + 4.34e12i)16-s − 1.18e13·17-s + ⋯
L(s)  = 1  + (−0.938 − 0.346i)2-s + 1.28i·3-s + (0.759 + 0.650i)4-s − 0.211i·5-s + (0.444 − 1.20i)6-s − 1.29·7-s + (−0.487 − 0.873i)8-s − 0.649·9-s + (−0.0731 + 0.198i)10-s + 0.305i·11-s + (−0.834 + 0.975i)12-s + 0.271i·13-s + (1.21 + 0.448i)14-s + 0.271·15-s + (0.154 + 0.987i)16-s − 1.42·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.487 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.487 + 0.873i$
Analytic conductor: \(22.3581\)
Root analytic conductor: \(4.72844\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :21/2),\ 0.487 + 0.873i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.446048 - 0.261780i\)
\(L(\frac12)\) \(\approx\) \(0.446048 - 0.261780i\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35e3 + 501. i)T \)
good3 \( 1 - 1.31e5iT - 1.04e10T^{2} \)
5 \( 1 + 4.61e6iT - 4.76e14T^{2} \)
7 \( 1 + 9.67e8T + 5.58e17T^{2} \)
11 \( 1 - 2.63e10iT - 7.40e21T^{2} \)
13 \( 1 - 1.34e11iT - 2.47e23T^{2} \)
17 \( 1 + 1.18e13T + 6.90e25T^{2} \)
19 \( 1 + 3.07e13iT - 7.14e26T^{2} \)
23 \( 1 - 5.37e12T + 3.94e28T^{2} \)
29 \( 1 + 2.15e15iT - 5.13e30T^{2} \)
31 \( 1 - 6.82e15T + 2.08e31T^{2} \)
37 \( 1 + 5.49e16iT - 8.55e32T^{2} \)
41 \( 1 + 1.39e16T + 7.38e33T^{2} \)
43 \( 1 - 2.25e17iT - 2.00e34T^{2} \)
47 \( 1 - 5.18e17T + 1.30e35T^{2} \)
53 \( 1 + 8.95e17iT - 1.62e36T^{2} \)
59 \( 1 + 7.68e18iT - 1.54e37T^{2} \)
61 \( 1 - 9.46e16iT - 3.10e37T^{2} \)
67 \( 1 - 8.03e18iT - 2.22e38T^{2} \)
71 \( 1 - 1.77e19T + 7.52e38T^{2} \)
73 \( 1 + 2.77e19T + 1.34e39T^{2} \)
79 \( 1 + 6.72e19T + 7.08e39T^{2} \)
83 \( 1 + 7.06e19iT - 1.99e40T^{2} \)
89 \( 1 + 1.26e20T + 8.65e40T^{2} \)
97 \( 1 + 9.20e20T + 5.27e41T^{2} \)
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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

−16.14949800720301407660813434917, −15.48795268062006304161661842488, −12.91308650611743895567158240884, −11.08287931453859709764366324638, −9.799444185455617570761739815496, −8.973597088974473422463162617439, −6.69703301800933309432306106043, −4.26272493611841040502188663214, −2.71036435983407830134486956191, −0.28417406310926938048550098896, 1.07273574754710418022754675014, 2.68205239510888456260534902999, 6.21270196242103732243909741417, 7.04736422082347176328421471968, 8.616813994826691736447610606616, 10.33544604394489013435148291433, 12.19368649900640974641532929640, 13.60980737803544064696759197958, 15.53634913980191163172572636008, 16.95137754900031093287358366058

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