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Properties

Label 2-2e3-8.5-c7-0-0
Degree $2$
Conductor $8$
Sign $0.0484 - 0.998i$
Analytic cond. $2.49908$
Root an. cond. $1.58084$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.70 − 5.81i)2-s + 40.2i·3-s + (60.3 + 112. i)4-s + 324. i·5-s + (233. − 390. i)6-s − 956.·7-s + (70.1 − 1.44e3i)8-s + 569.·9-s + (1.88e3 − 3.14e3i)10-s + 5.45e3i·11-s + (−4.53e3 + 2.42e3i)12-s − 6.28e3i·13-s + (9.28e3 + 5.56e3i)14-s − 1.30e4·15-s + (−9.09e3 + 1.36e4i)16-s + 3.45e4·17-s + ⋯
L(s)  = 1  + (−0.857 − 0.513i)2-s + 0.859i·3-s + (0.471 + 0.881i)4-s + 1.16i·5-s + (0.441 − 0.737i)6-s − 1.05·7-s + (0.0484 − 0.998i)8-s + 0.260·9-s + (0.596 − 0.995i)10-s + 1.23i·11-s + (−0.758 + 0.405i)12-s − 0.793i·13-s + (0.904 + 0.541i)14-s − 0.998·15-s + (−0.554 + 0.831i)16-s + 1.70·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0484 - 0.998i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0484 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.0484 - 0.998i$
Analytic conductor: \(2.49908\)
Root analytic conductor: \(1.58084\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :7/2),\ 0.0484 - 0.998i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.571060 + 0.544050i\)
\(L(\frac12)\) \(\approx\) \(0.571060 + 0.544050i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (9.70 + 5.81i)T \)
good3 \( 1 - 40.2iT - 2.18e3T^{2} \)
5 \( 1 - 324. iT - 7.81e4T^{2} \)
7 \( 1 + 956.T + 8.23e5T^{2} \)
11 \( 1 - 5.45e3iT - 1.94e7T^{2} \)
13 \( 1 + 6.28e3iT - 6.27e7T^{2} \)
17 \( 1 - 3.45e4T + 4.10e8T^{2} \)
19 \( 1 + 1.45e4iT - 8.93e8T^{2} \)
23 \( 1 + 2.46e4T + 3.40e9T^{2} \)
29 \( 1 - 1.71e5iT - 1.72e10T^{2} \)
31 \( 1 - 1.11e5T + 2.75e10T^{2} \)
37 \( 1 + 1.03e5iT - 9.49e10T^{2} \)
41 \( 1 - 7.16e4T + 1.94e11T^{2} \)
43 \( 1 + 3.28e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.19e5T + 5.06e11T^{2} \)
53 \( 1 + 1.04e6iT - 1.17e12T^{2} \)
59 \( 1 + 2.25e5iT - 2.48e12T^{2} \)
61 \( 1 - 1.55e6iT - 3.14e12T^{2} \)
67 \( 1 - 3.16e5iT - 6.06e12T^{2} \)
71 \( 1 - 5.38e5T + 9.09e12T^{2} \)
73 \( 1 + 2.68e6T + 1.10e13T^{2} \)
79 \( 1 - 8.22e6T + 1.92e13T^{2} \)
83 \( 1 + 5.89e6iT - 2.71e13T^{2} \)
89 \( 1 - 4.37e5T + 4.42e13T^{2} \)
97 \( 1 + 7.84e6T + 8.07e13T^{2} \)
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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

−20.54976960222706459396564574709, −19.17123510287404719411735124174, −17.96657354667578831454629839509, −16.27537246410445903238815811695, −15.05692921359589120085438163400, −12.52018765799442341814866117163, −10.44905461275987625856122430124, −9.773344448429515629654896213894, −7.15828558850434337911417394281, −3.26523589355173736921457070848, 0.933573956135023573850337573008, 6.14138368689667803508768059210, 8.052360778405481240336541638938, 9.674809299339302609644667991477, 12.16219370510172461389524733756, 13.73241045867163548595729855476, 16.10170398260580703930751894807, 16.82893870980081922120234998536, 18.74296398959104829030844644836, 19.34674845977546107917432005342

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