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Properties

Label 2-2e3-8.5-c21-0-4
Degree $2$
Conductor $8$
Sign $-0.711 - 0.702i$
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  + (−377. − 1.39e3i)2-s + 1.48e5i·3-s + (−1.81e6 + 1.05e6i)4-s + 2.51e7i·5-s + (2.07e8 − 5.60e7i)6-s + 8.96e8·7-s + (2.16e9 + 2.13e9i)8-s − 1.15e10·9-s + (3.51e10 − 9.51e9i)10-s + 1.46e11i·11-s + (−1.56e11 − 2.68e11i)12-s − 4.53e11i·13-s + (−3.38e11 − 1.25e12i)14-s − 3.73e12·15-s + (2.16e12 − 3.82e12i)16-s + 6.08e11·17-s + ⋯
L(s)  = 1  + (−0.260 − 0.965i)2-s + 1.44i·3-s + (−0.863 + 0.503i)4-s + 1.15i·5-s + (1.39 − 0.378i)6-s + 1.19·7-s + (0.711 + 0.702i)8-s − 1.10·9-s + (1.11 − 0.300i)10-s + 1.70i·11-s + (−0.730 − 1.25i)12-s − 0.913i·13-s + (−0.312 − 1.15i)14-s − 1.67·15-s + (0.492 − 0.870i)16-s + 0.0731·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.711 - 0.702i$
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: \((2,\ 8,\ (\ :21/2),\ -0.711 - 0.702i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.534129 + 1.30144i\)
\(L(\frac12)\) \(\approx\) \(0.534129 + 1.30144i\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (377. + 1.39e3i)T \)
good3 \( 1 - 1.48e5iT - 1.04e10T^{2} \)
5 \( 1 - 2.51e7iT - 4.76e14T^{2} \)
7 \( 1 - 8.96e8T + 5.58e17T^{2} \)
11 \( 1 - 1.46e11iT - 7.40e21T^{2} \)
13 \( 1 + 4.53e11iT - 2.47e23T^{2} \)
17 \( 1 - 6.08e11T + 6.90e25T^{2} \)
19 \( 1 - 3.33e13iT - 7.14e26T^{2} \)
23 \( 1 + 2.43e14T + 3.94e28T^{2} \)
29 \( 1 + 1.19e15iT - 5.13e30T^{2} \)
31 \( 1 - 5.18e15T + 2.08e31T^{2} \)
37 \( 1 + 2.87e15iT - 8.55e32T^{2} \)
41 \( 1 + 9.95e16T + 7.38e33T^{2} \)
43 \( 1 + 2.13e17iT - 2.00e34T^{2} \)
47 \( 1 + 2.65e17T + 1.30e35T^{2} \)
53 \( 1 - 1.06e17iT - 1.62e36T^{2} \)
59 \( 1 + 3.03e18iT - 1.54e37T^{2} \)
61 \( 1 - 3.77e18iT - 3.10e37T^{2} \)
67 \( 1 + 7.43e18iT - 2.22e38T^{2} \)
71 \( 1 - 3.77e19T + 7.52e38T^{2} \)
73 \( 1 + 3.15e19T + 1.34e39T^{2} \)
79 \( 1 - 8.13e19T + 7.08e39T^{2} \)
83 \( 1 - 1.42e20iT - 1.99e40T^{2} \)
89 \( 1 - 4.63e20T + 8.65e40T^{2} \)
97 \( 1 - 8.42e20T + 5.27e41T^{2} \)
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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

−17.35053626553846005425032150730, −15.24744907261667675131537168487, −14.34948756943896720428099931952, −11.93398903310409163070593766942, −10.49293515509575645531244309965, −9.977248999933452415735374381166, −7.950301912108482844872567295056, −4.91203877597456706995979696593, −3.69627357905105783200711277833, −2.06784220137434101707282660143, 0.57737106757040686956272165596, 1.48704853612403231827358110360, 4.84682875667856006004387443881, 6.35341574420010375996231469120, 7.992387204579399978058872651912, 8.731227500130108269458442342233, 11.61537604353753477951574648885, 13.28840660758551046183601393823, 14.12598163215184906543001957828, 16.23353281107151424432216336230

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