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Properties

Label 2-2e3-8.3-c12-0-8
Degree $2$
Conductor $8$
Sign $0.984 + 0.173i$
Analytic cond. $7.31195$
Root an. cond. $2.70406$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−35.1 + 53.4i)2-s + 1.27e3·3-s + (−1.62e3 − 3.76e3i)4-s − 1.90e4i·5-s + (−4.47e4 + 6.80e4i)6-s − 1.36e5i·7-s + (2.58e5 + 4.55e4i)8-s + 1.08e6·9-s + (1.02e6 + 6.71e5i)10-s − 4.78e5·11-s + (−2.06e6 − 4.78e6i)12-s + 4.87e6i·13-s + (7.28e6 + 4.78e6i)14-s − 2.42e7i·15-s + (−1.15e7 + 1.22e7i)16-s + 9.96e6·17-s + ⋯
L(s)  = 1  + (−0.549 + 0.835i)2-s + 1.74·3-s + (−0.395 − 0.918i)4-s − 1.22i·5-s + (−0.959 + 1.45i)6-s − 1.15i·7-s + (0.984 + 0.173i)8-s + 2.04·9-s + (1.02 + 0.671i)10-s − 0.270·11-s + (−0.690 − 1.60i)12-s + 1.01i·13-s + (0.966 + 0.636i)14-s − 2.13i·15-s + (−0.686 + 0.727i)16-s + 0.412·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(7.31195\)
Root analytic conductor: \(2.70406\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :6),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.05994 - 0.180444i\)
\(L(\frac12)\) \(\approx\) \(2.05994 - 0.180444i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (35.1 - 53.4i)T \)
good3 \( 1 - 1.27e3T + 5.31e5T^{2} \)
5 \( 1 + 1.90e4iT - 2.44e8T^{2} \)
7 \( 1 + 1.36e5iT - 1.38e10T^{2} \)
11 \( 1 + 4.78e5T + 3.13e12T^{2} \)
13 \( 1 - 4.87e6iT - 2.32e13T^{2} \)
17 \( 1 - 9.96e6T + 5.82e14T^{2} \)
19 \( 1 - 8.47e5T + 2.21e15T^{2} \)
23 \( 1 - 1.02e8iT - 2.19e16T^{2} \)
29 \( 1 + 5.58e8iT - 3.53e17T^{2} \)
31 \( 1 - 4.80e8iT - 7.87e17T^{2} \)
37 \( 1 - 2.73e9iT - 6.58e18T^{2} \)
41 \( 1 + 4.60e9T + 2.25e19T^{2} \)
43 \( 1 - 1.16e10T + 3.99e19T^{2} \)
47 \( 1 - 5.60e9iT - 1.16e20T^{2} \)
53 \( 1 + 1.34e10iT - 4.91e20T^{2} \)
59 \( 1 + 5.52e10T + 1.77e21T^{2} \)
61 \( 1 - 4.88e10iT - 2.65e21T^{2} \)
67 \( 1 - 4.34e10T + 8.18e21T^{2} \)
71 \( 1 - 2.13e11iT - 1.64e22T^{2} \)
73 \( 1 + 1.51e11T + 2.29e22T^{2} \)
79 \( 1 + 8.43e10iT - 5.90e22T^{2} \)
83 \( 1 + 1.96e11T + 1.06e23T^{2} \)
89 \( 1 + 4.31e11T + 2.46e23T^{2} \)
97 \( 1 - 1.63e12T + 6.93e23T^{2} \)
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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

−19.02347481722619184657317126000, −17.01572775848817607161837135536, −15.79420405208233423916525843407, −14.20731506387789770407019838173, −13.33230372587284571841881378673, −9.806626552870910899652773481654, −8.647080204474068762157103122100, −7.44168847884058486974420620773, −4.33047987072581656482729135933, −1.31752065089440526798832245761, 2.35140364477098046879374861839, 3.22803982532504872327437878837, 7.68260323387344551988171478414, 9.009726883036528062374512904991, 10.48025721390247747807617190605, 12.65491368034762285397868592339, 14.27283974852143273699539133301, 15.44345471497832134196167876185, 18.27756103394844969447594355673, 18.87284879976070244756686325696

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