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Properties

Label 2-2e3-8.5-c13-0-2
Degree $2$
Conductor $8$
Sign $0.367 - 0.930i$
Analytic cond. $8.57847$
Root an. cond. $2.92890$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−83.4 + 35.1i)2-s + 622. i·3-s + (5.72e3 − 5.85e3i)4-s − 2.66e4i·5-s + (−2.18e4 − 5.19e4i)6-s + 5.04e4·7-s + (−2.72e5 + 6.89e5i)8-s + 1.20e6·9-s + (9.36e5 + 2.22e6i)10-s + 9.06e6i·11-s + (3.64e6 + 3.56e6i)12-s + 8.06e6i·13-s + (−4.21e6 + 1.77e6i)14-s + 1.65e7·15-s + (−1.49e6 − 6.70e7i)16-s + 5.68e7·17-s + ⋯
L(s)  = 1  + (−0.921 + 0.387i)2-s + 0.492i·3-s + (0.699 − 0.714i)4-s − 0.763i·5-s + (−0.191 − 0.454i)6-s + 0.162·7-s + (−0.367 + 0.930i)8-s + 0.757·9-s + (0.296 + 0.703i)10-s + 1.54i·11-s + (0.352 + 0.344i)12-s + 0.463i·13-s + (−0.149 + 0.0628i)14-s + 0.376·15-s + (−0.0222 − 0.999i)16-s + 0.570·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.367 - 0.930i$
Analytic conductor: \(8.57847\)
Root analytic conductor: \(2.92890\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :13/2),\ 0.367 - 0.930i)\)

Particular Values

\(L(7)\) \(\approx\) \(0.958609 + 0.652172i\)
\(L(\frac12)\) \(\approx\) \(0.958609 + 0.652172i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (83.4 - 35.1i)T \)
good3 \( 1 - 622. iT - 1.59e6T^{2} \)
5 \( 1 + 2.66e4iT - 1.22e9T^{2} \)
7 \( 1 - 5.04e4T + 9.68e10T^{2} \)
11 \( 1 - 9.06e6iT - 3.45e13T^{2} \)
13 \( 1 - 8.06e6iT - 3.02e14T^{2} \)
17 \( 1 - 5.68e7T + 9.90e15T^{2} \)
19 \( 1 - 1.06e8iT - 4.20e16T^{2} \)
23 \( 1 - 6.31e8T + 5.04e17T^{2} \)
29 \( 1 - 4.06e9iT - 1.02e19T^{2} \)
31 \( 1 - 5.86e9T + 2.44e19T^{2} \)
37 \( 1 + 2.97e10iT - 2.43e20T^{2} \)
41 \( 1 + 5.32e10T + 9.25e20T^{2} \)
43 \( 1 - 8.32e9iT - 1.71e21T^{2} \)
47 \( 1 - 1.06e11T + 5.46e21T^{2} \)
53 \( 1 - 1.04e11iT - 2.60e22T^{2} \)
59 \( 1 - 3.49e11iT - 1.04e23T^{2} \)
61 \( 1 + 2.73e11iT - 1.61e23T^{2} \)
67 \( 1 + 4.69e11iT - 5.48e23T^{2} \)
71 \( 1 + 4.62e11T + 1.16e24T^{2} \)
73 \( 1 - 4.02e11T + 1.67e24T^{2} \)
79 \( 1 + 3.34e12T + 4.66e24T^{2} \)
83 \( 1 + 4.78e12iT - 8.87e24T^{2} \)
89 \( 1 - 3.03e12T + 2.19e25T^{2} \)
97 \( 1 - 4.70e12T + 6.73e25T^{2} \)
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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

−18.61435859747409530483747622443, −17.19094015933675005011469975264, −16.04039282456000192846248010801, −14.79371855526870094959824620386, −12.37522726435827470647074218513, −10.31154564380433380534475914945, −9.088730569669392619283594539756, −7.23025262144818735920111066551, −4.84951984004623786832158653480, −1.46182439319606293728102147730, 0.916042999419600349163614330458, 3.00011105243157225660093515947, 6.66886947381929636598797455366, 8.219003349432397723867459204856, 10.23647161746139592069125326907, 11.59294430572957062362088139588, 13.38308672546338184734972988077, 15.49015883075795682327378869653, 17.10832439975549334925335580040, 18.61753644560543499677374162446

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