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Properties

Label 2-2e3-8.5-c21-0-19
Degree $2$
Conductor $8$
Sign $0.457 - 0.889i$
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  + (228. − 1.43e3i)2-s − 1.81e5i·3-s + (−1.99e6 − 6.52e5i)4-s − 6.89e6i·5-s + (−2.59e8 − 4.14e7i)6-s + 1.45e8·7-s + (−1.38e9 + 2.70e9i)8-s − 2.24e10·9-s + (−9.85e9 − 1.57e9i)10-s + 9.88e10i·11-s + (−1.18e11 + 3.61e11i)12-s − 7.50e11i·13-s + (3.32e10 − 2.08e11i)14-s − 1.25e12·15-s + (3.54e12 + 2.60e12i)16-s − 4.40e12·17-s + ⋯
L(s)  = 1  + (0.157 − 0.987i)2-s − 1.77i·3-s + (−0.950 − 0.311i)4-s − 0.315i·5-s + (−1.75 − 0.279i)6-s + 0.195·7-s + (−0.457 + 0.889i)8-s − 2.15·9-s + (−0.311 − 0.0497i)10-s + 1.14i·11-s + (−0.552 + 1.68i)12-s − 1.50i·13-s + (0.0307 − 0.192i)14-s − 0.560·15-s + (0.806 + 0.591i)16-s − 0.529·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.457 - 0.889i$
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: \((2,\ 8,\ (\ :21/2),\ 0.457 - 0.889i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.751432 + 0.458686i\)
\(L(\frac12)\) \(\approx\) \(0.751432 + 0.458686i\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-228. + 1.43e3i)T \)
good3 \( 1 + 1.81e5iT - 1.04e10T^{2} \)
5 \( 1 + 6.89e6iT - 4.76e14T^{2} \)
7 \( 1 - 1.45e8T + 5.58e17T^{2} \)
11 \( 1 - 9.88e10iT - 7.40e21T^{2} \)
13 \( 1 + 7.50e11iT - 2.47e23T^{2} \)
17 \( 1 + 4.40e12T + 6.90e25T^{2} \)
19 \( 1 + 3.54e13iT - 7.14e26T^{2} \)
23 \( 1 - 5.59e13T + 3.94e28T^{2} \)
29 \( 1 - 4.01e15iT - 5.13e30T^{2} \)
31 \( 1 + 5.34e15T + 2.08e31T^{2} \)
37 \( 1 + 4.56e16iT - 8.55e32T^{2} \)
41 \( 1 - 2.74e16T + 7.38e33T^{2} \)
43 \( 1 + 3.58e16iT - 2.00e34T^{2} \)
47 \( 1 + 2.35e17T + 1.30e35T^{2} \)
53 \( 1 - 2.77e17iT - 1.62e36T^{2} \)
59 \( 1 + 2.39e18iT - 1.54e37T^{2} \)
61 \( 1 - 6.56e18iT - 3.10e37T^{2} \)
67 \( 1 - 1.37e19iT - 2.22e38T^{2} \)
71 \( 1 + 1.35e19T + 7.52e38T^{2} \)
73 \( 1 - 3.67e19T + 1.34e39T^{2} \)
79 \( 1 + 1.38e20T + 7.08e39T^{2} \)
83 \( 1 + 9.59e17iT - 1.99e40T^{2} \)
89 \( 1 - 7.87e19T + 8.65e40T^{2} \)
97 \( 1 + 8.67e20T + 5.27e41T^{2} \)
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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

−14.63395715046781173564727475096, −12.96676540062708617816044829237, −12.63526953155318994984812825133, −11.00897028306926944472941968295, −8.800578863080740612285215267079, −7.23306899372500626307669788083, −5.21794923976714985048722113574, −2.73009393070821266386583630782, −1.47477423946508531446286441788, −0.30073062795388891553409005998, 3.50658771725605611021003369146, 4.64587210760454883781946917964, 6.14893187606471071453431061910, 8.488356994016258244577426530917, 9.684610064173417386051166860258, 11.28968564542712821823721277318, 13.97147900510504956332424415641, 14.91651246627839089255740031555, 16.21750763161918950611190813503, 16.86967275966956567554359323466

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