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Properties

Label 2-2e3-8.3-c10-0-2
Degree $2$
Conductor $8$
Sign $-0.118 - 0.992i$
Analytic cond. $5.08285$
Root an. cond. $2.25451$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−27.0 + 17.0i)2-s + 203.·3-s + (440. − 924. i)4-s + 2.08e3i·5-s + (−5.51e3 + 3.48e3i)6-s + 2.53e4i·7-s + (3.87e3 + 3.25e4i)8-s − 1.74e4·9-s + (−3.56e4 − 5.65e4i)10-s + 1.32e5·11-s + (8.97e4 − 1.88e5i)12-s + 3.04e5i·13-s + (−4.33e5 − 6.86e5i)14-s + 4.25e5i·15-s + (−6.60e5 − 8.14e5i)16-s + 1.87e6·17-s + ⋯
L(s)  = 1  + (−0.845 + 0.533i)2-s + 0.838·3-s + (0.430 − 0.902i)4-s + 0.668i·5-s + (−0.709 + 0.447i)6-s + 1.50i·7-s + (0.118 + 0.992i)8-s − 0.296·9-s + (−0.356 − 0.565i)10-s + 0.822·11-s + (0.360 − 0.757i)12-s + 0.818i·13-s + (−0.805 − 1.27i)14-s + 0.560i·15-s + (−0.630 − 0.776i)16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.118 - 0.992i$
Analytic conductor: \(5.08285\)
Root analytic conductor: \(2.25451\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :5),\ -0.118 - 0.992i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.846644 + 0.953498i\)
\(L(\frac12)\) \(\approx\) \(0.846644 + 0.953498i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (27.0 - 17.0i)T \)
good3 \( 1 - 203.T + 5.90e4T^{2} \)
5 \( 1 - 2.08e3iT - 9.76e6T^{2} \)
7 \( 1 - 2.53e4iT - 2.82e8T^{2} \)
11 \( 1 - 1.32e5T + 2.59e10T^{2} \)
13 \( 1 - 3.04e5iT - 1.37e11T^{2} \)
17 \( 1 - 1.87e6T + 2.01e12T^{2} \)
19 \( 1 + 2.20e6T + 6.13e12T^{2} \)
23 \( 1 + 4.83e6iT - 4.14e13T^{2} \)
29 \( 1 + 2.08e7iT - 4.20e14T^{2} \)
31 \( 1 + 4.27e7iT - 8.19e14T^{2} \)
37 \( 1 - 1.24e8iT - 4.80e15T^{2} \)
41 \( 1 - 1.24e8T + 1.34e16T^{2} \)
43 \( 1 - 3.15e7T + 2.16e16T^{2} \)
47 \( 1 - 2.91e7iT - 5.25e16T^{2} \)
53 \( 1 + 5.07e8iT - 1.74e17T^{2} \)
59 \( 1 - 1.64e8T + 5.11e17T^{2} \)
61 \( 1 - 3.21e7iT - 7.13e17T^{2} \)
67 \( 1 + 3.70e8T + 1.82e18T^{2} \)
71 \( 1 - 3.20e8iT - 3.25e18T^{2} \)
73 \( 1 - 2.69e9T + 4.29e18T^{2} \)
79 \( 1 - 1.89e9iT - 9.46e18T^{2} \)
83 \( 1 + 2.20e9T + 1.55e19T^{2} \)
89 \( 1 - 9.12e9T + 3.11e19T^{2} \)
97 \( 1 + 1.89e9T + 7.37e19T^{2} \)
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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

−19.18871265505293456054372572018, −18.68836751623296679802979352670, −16.86819266171245616122630891253, −15.07083007715782273253122454037, −14.41935146110408586650329121104, −11.63631272004222429945714160926, −9.491726686868757772705355187366, −8.303289285099009083116945148472, −6.21673432889560886238719746609, −2.39897984384225522561967519549, 1.03419648119856735476422706847, 3.56545403582249143717818311979, 7.57975177776673887087490715174, 9.004328092558193610240052100303, 10.62189975802652944437665526456, 12.64697616339734012808282389767, 14.25285047539415652996728684351, 16.49340405379745522609209122535, 17.45655309799645770692031099885, 19.53212226255706675326980883147

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