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Properties

Label 2-2e3-8.5-c21-0-7
Degree $2$
Conductor $8$
Sign $0.299 - 0.954i$
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  + (1.17e3 + 847. i)2-s − 1.78e5i·3-s + (6.61e5 + 1.98e6i)4-s + 3.12e7i·5-s + (1.51e8 − 2.09e8i)6-s + 2.98e8·7-s + (−9.08e8 + 2.89e9i)8-s − 2.14e10·9-s + (−2.64e10 + 3.67e10i)10-s − 1.69e10i·11-s + (3.55e11 − 1.18e11i)12-s + 5.06e11i·13-s + (3.51e11 + 2.53e11i)14-s + 5.58e12·15-s + (−3.52e12 + 2.63e12i)16-s + 9.27e12·17-s + ⋯
L(s)  = 1  + (0.811 + 0.584i)2-s − 1.74i·3-s + (0.315 + 0.948i)4-s + 1.43i·5-s + (1.02 − 1.41i)6-s + 0.399·7-s + (−0.299 + 0.954i)8-s − 2.05·9-s + (−0.837 + 1.16i)10-s − 0.196i·11-s + (1.65 − 0.551i)12-s + 1.01i·13-s + (0.324 + 0.233i)14-s + 2.50·15-s + (−0.800 + 0.598i)16-s + 1.11·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.299 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.299 - 0.954i$
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: \((2,\ 8,\ (\ :21/2),\ 0.299 - 0.954i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.30225 + 1.69108i\)
\(L(\frac12)\) \(\approx\) \(2.30225 + 1.69108i\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17e3 - 847. i)T \)
good3 \( 1 + 1.78e5iT - 1.04e10T^{2} \)
5 \( 1 - 3.12e7iT - 4.76e14T^{2} \)
7 \( 1 - 2.98e8T + 5.58e17T^{2} \)
11 \( 1 + 1.69e10iT - 7.40e21T^{2} \)
13 \( 1 - 5.06e11iT - 2.47e23T^{2} \)
17 \( 1 - 9.27e12T + 6.90e25T^{2} \)
19 \( 1 - 5.18e13iT - 7.14e26T^{2} \)
23 \( 1 - 2.52e14T + 3.94e28T^{2} \)
29 \( 1 + 3.59e14iT - 5.13e30T^{2} \)
31 \( 1 - 7.41e14T + 2.08e31T^{2} \)
37 \( 1 + 3.56e16iT - 8.55e32T^{2} \)
41 \( 1 + 4.38e16T + 7.38e33T^{2} \)
43 \( 1 - 6.67e16iT - 2.00e34T^{2} \)
47 \( 1 + 1.54e17T + 1.30e35T^{2} \)
53 \( 1 - 1.64e18iT - 1.62e36T^{2} \)
59 \( 1 + 4.37e18iT - 1.54e37T^{2} \)
61 \( 1 - 3.59e17iT - 3.10e37T^{2} \)
67 \( 1 + 1.04e19iT - 2.22e38T^{2} \)
71 \( 1 - 2.52e19T + 7.52e38T^{2} \)
73 \( 1 + 9.66e18T + 1.34e39T^{2} \)
79 \( 1 + 5.47e19T + 7.08e39T^{2} \)
83 \( 1 - 1.31e20iT - 1.99e40T^{2} \)
89 \( 1 + 2.00e20T + 8.65e40T^{2} \)
97 \( 1 - 1.40e21T + 5.27e41T^{2} \)
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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

−16.88114159315477634543616170509, −14.54872268943671226544851350974, −14.04742443051236468274670694276, −12.46081598890978418836745472696, −11.31141439477169113325007126699, −7.949061474677954028126466419040, −6.99159942664620199599365424116, −5.95174918653611339732608547534, −3.21534349295862331366456535261, −1.79807627331119542008590493688, 0.78390255049273007730040633420, 3.17156184786151191438899533521, 4.76297274304642844193177515423, 5.19774308542985226974402675659, 8.868746480560423340584468994065, 10.09617749302770534695643821564, 11.48250711693640019042062989988, 13.09342140591813658847899917967, 14.88567227066064373770295822320, 15.81983804234260215623435672693

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