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Properties

Label 2-2e3-8.5-c21-0-6
Degree $2$
Conductor $8$
Sign $-0.976 - 0.215i$
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  + (1.44e3 + 104. i)2-s + 1.16e5i·3-s + (2.07e6 + 3.02e5i)4-s + 3.48e7i·5-s + (−1.21e7 + 1.68e8i)6-s − 1.26e9·7-s + (2.96e9 + 6.54e8i)8-s − 3.08e9·9-s + (−3.65e9 + 5.04e10i)10-s − 3.61e10i·11-s + (−3.52e10 + 2.41e11i)12-s − 5.83e11i·13-s + (−1.82e12 − 1.32e11i)14-s − 4.06e12·15-s + (4.21e12 + 1.25e12i)16-s + 3.71e12·17-s + ⋯
L(s)  = 1  + (0.997 + 0.0723i)2-s + 1.13i·3-s + (0.989 + 0.144i)4-s + 1.59i·5-s + (−0.0823 + 1.13i)6-s − 1.69·7-s + (0.976 + 0.215i)8-s − 0.294·9-s + (−0.115 + 1.59i)10-s − 0.420i·11-s + (−0.164 + 1.12i)12-s − 1.17i·13-s + (−1.68 − 0.122i)14-s − 1.81·15-s + (0.958 + 0.285i)16-s + 0.446·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.976 - 0.215i$
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: \((2,\ 8,\ (\ :21/2),\ -0.976 - 0.215i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.292048 + 2.67855i\)
\(L(\frac12)\) \(\approx\) \(0.292048 + 2.67855i\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.44e3 - 104. i)T \)
good3 \( 1 - 1.16e5iT - 1.04e10T^{2} \)
5 \( 1 - 3.48e7iT - 4.76e14T^{2} \)
7 \( 1 + 1.26e9T + 5.58e17T^{2} \)
11 \( 1 + 3.61e10iT - 7.40e21T^{2} \)
13 \( 1 + 5.83e11iT - 2.47e23T^{2} \)
17 \( 1 - 3.71e12T + 6.90e25T^{2} \)
19 \( 1 - 1.39e13iT - 7.14e26T^{2} \)
23 \( 1 - 6.23e12T + 3.94e28T^{2} \)
29 \( 1 - 3.96e15iT - 5.13e30T^{2} \)
31 \( 1 + 5.98e15T + 2.08e31T^{2} \)
37 \( 1 - 3.67e16iT - 8.55e32T^{2} \)
41 \( 1 - 4.32e16T + 7.38e33T^{2} \)
43 \( 1 - 1.55e16iT - 2.00e34T^{2} \)
47 \( 1 - 2.82e17T + 1.30e35T^{2} \)
53 \( 1 + 3.04e17iT - 1.62e36T^{2} \)
59 \( 1 + 2.15e18iT - 1.54e37T^{2} \)
61 \( 1 + 4.15e18iT - 3.10e37T^{2} \)
67 \( 1 - 2.12e19iT - 2.22e38T^{2} \)
71 \( 1 - 1.05e19T + 7.52e38T^{2} \)
73 \( 1 - 1.19e19T + 1.34e39T^{2} \)
79 \( 1 + 1.10e20T + 7.08e39T^{2} \)
83 \( 1 + 4.07e19iT - 1.99e40T^{2} \)
89 \( 1 - 3.12e20T + 8.65e40T^{2} \)
97 \( 1 + 3.40e20T + 5.27e41T^{2} \)
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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

−16.40638588481784803707552521914, −15.48642775628588432017012342546, −14.46455155822741171097320672789, −12.82493768950023868038972939809, −10.80168287271634385100764380656, −10.03067609103483930874453986695, −7.03320840709020172874275193070, −5.73535423300282259265612922343, −3.50028455387228854278702651741, −3.09195358037859800580735222119, 0.64946534521072341567170746477, 2.08084479300789977677156775734, 4.14939535343162971117248464314, 5.95895614886110468361338193573, 7.26195837866985155767764129414, 9.421841321082223361405097476269, 12.09956139538407709418751759477, 12.79879429599426985683328251953, 13.59221372187931751587562745786, 15.88124701665027150050216466684

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