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Properties

Label 2-2e3-8.3-c16-0-14
Degree $2$
Conductor $8$
Sign $-0.876 - 0.481i$
Analytic cond. $12.9859$
Root an. cond. $3.60360$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (163. − 197. i)2-s − 2.12e3·3-s + (−1.22e4 − 6.43e4i)4-s − 1.55e5i·5-s + (−3.46e5 + 4.18e5i)6-s + 8.29e5i·7-s + (−1.47e7 − 8.08e6i)8-s − 3.85e7·9-s + (−3.07e7 − 2.54e7i)10-s − 7.43e7·11-s + (2.60e7 + 1.36e8i)12-s + 6.40e8i·13-s + (1.63e8 + 1.35e8i)14-s + 3.30e8i·15-s + (−3.99e9 + 1.58e9i)16-s − 4.86e9·17-s + ⋯
L(s)  = 1  + (0.637 − 0.770i)2-s − 0.323·3-s + (−0.187 − 0.982i)4-s − 0.398i·5-s + (−0.206 + 0.249i)6-s + 0.143i·7-s + (−0.876 − 0.481i)8-s − 0.895·9-s + (−0.307 − 0.254i)10-s − 0.346·11-s + (0.0605 + 0.317i)12-s + 0.785i·13-s + (0.110 + 0.0917i)14-s + 0.128i·15-s + (−0.929 + 0.368i)16-s − 0.696·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.876 - 0.481i$
Analytic conductor: \(12.9859\)
Root analytic conductor: \(3.60360\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :8),\ -0.876 - 0.481i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.201064 + 0.782940i\)
\(L(\frac12)\) \(\approx\) \(0.201064 + 0.782940i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-163. + 197. i)T \)
good3 \( 1 + 2.12e3T + 4.30e7T^{2} \)
5 \( 1 + 1.55e5iT - 1.52e11T^{2} \)
7 \( 1 - 8.29e5iT - 3.32e13T^{2} \)
11 \( 1 + 7.43e7T + 4.59e16T^{2} \)
13 \( 1 - 6.40e8iT - 6.65e17T^{2} \)
17 \( 1 + 4.86e9T + 4.86e19T^{2} \)
19 \( 1 + 1.49e10T + 2.88e20T^{2} \)
23 \( 1 + 5.37e10iT - 6.13e21T^{2} \)
29 \( 1 + 5.66e11iT - 2.50e23T^{2} \)
31 \( 1 + 1.31e12iT - 7.27e23T^{2} \)
37 \( 1 + 2.80e12iT - 1.23e25T^{2} \)
41 \( 1 + 9.61e11T + 6.37e25T^{2} \)
43 \( 1 + 6.98e12T + 1.36e26T^{2} \)
47 \( 1 + 1.89e13iT - 5.66e26T^{2} \)
53 \( 1 - 8.89e13iT - 3.87e27T^{2} \)
59 \( 1 + 2.65e14T + 2.15e28T^{2} \)
61 \( 1 - 2.98e14iT - 3.67e28T^{2} \)
67 \( 1 + 5.63e13T + 1.64e29T^{2} \)
71 \( 1 + 1.07e15iT - 4.16e29T^{2} \)
73 \( 1 - 9.51e14T + 6.50e29T^{2} \)
79 \( 1 + 2.31e15iT - 2.30e30T^{2} \)
83 \( 1 + 1.49e15T + 5.07e30T^{2} \)
89 \( 1 + 1.19e15T + 1.54e31T^{2} \)
97 \( 1 + 5.34e15T + 6.14e31T^{2} \)
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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

−16.86147996748600061361891816274, −15.04060202856811369457381387483, −13.53402598901111497287889493580, −12.09451744826852663194912895233, −10.83531984608322213916938492790, −8.947690403359466465745612722145, −6.08549071209459623488354467492, −4.46775759211411115719007850674, −2.35026911402547793720189193262, −0.27332495862600666766722802676, 3.06897568943048632659149792715, 5.15464238876864772384304667699, 6.69207267382040999204456115858, 8.459890815063787775069574383369, 10.96604720938603539530167167085, 12.67458500665388728919750822415, 14.18871935444066161335698123194, 15.46791549722930217674158521430, 16.96117242495258759426454240094, 18.05796587738779963764320864457

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