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Properties

Label 2-2e3-8.3-c18-0-1
Degree $2$
Conductor $8$
Sign $-0.000287 + 0.999i$
Analytic cond. $16.4308$
Root an. cond. $4.05350$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0491 + 511. i)2-s − 9.97e3·3-s + (−2.62e5 − 50.3i)4-s + 3.00e6i·5-s + (490. − 5.10e6i)6-s + 5.44e7i·7-s + (3.86e4 − 1.34e8i)8-s − 2.87e8·9-s + (−1.53e9 − 1.47e5i)10-s + 1.04e8·11-s + (2.61e9 + 5.02e5i)12-s − 4.11e9i·13-s + (−2.78e10 − 2.67e6i)14-s − 2.99e10i·15-s + (6.87e10 + 2.63e7i)16-s + 2.03e11·17-s + ⋯
L(s)  = 1  + (−9.59e−5 + 0.999i)2-s − 0.506·3-s + (−0.999 − 0.000191i)4-s + 1.53i·5-s + (4.86e−5 − 0.506i)6-s + 1.34i·7-s + (0.000287 − 0.999i)8-s − 0.742·9-s + (−1.53 − 0.000147i)10-s + 0.0441·11-s + (0.506 + 9.72e−5i)12-s − 0.388i·13-s + (−1.34 − 0.000129i)14-s − 0.780i·15-s + (0.999 + 0.000383i)16-s + 1.71·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000287 + 0.999i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.000287 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.000287 + 0.999i$
Analytic conductor: \(16.4308\)
Root analytic conductor: \(4.05350\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :9),\ -0.000287 + 0.999i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.381361 - 0.381471i\)
\(L(\frac12)\) \(\approx\) \(0.381361 - 0.381471i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0491 - 511. i)T \)
good3 \( 1 + 9.97e3T + 3.87e8T^{2} \)
5 \( 1 - 3.00e6iT - 3.81e12T^{2} \)
7 \( 1 - 5.44e7iT - 1.62e15T^{2} \)
11 \( 1 - 1.04e8T + 5.55e18T^{2} \)
13 \( 1 + 4.11e9iT - 1.12e20T^{2} \)
17 \( 1 - 2.03e11T + 1.40e22T^{2} \)
19 \( 1 + 4.23e11T + 1.04e23T^{2} \)
23 \( 1 + 2.35e12iT - 3.24e24T^{2} \)
29 \( 1 - 1.21e13iT - 2.10e26T^{2} \)
31 \( 1 - 4.32e12iT - 6.99e26T^{2} \)
37 \( 1 - 6.48e13iT - 1.68e28T^{2} \)
41 \( 1 + 4.74e14T + 1.07e29T^{2} \)
43 \( 1 + 9.30e13T + 2.52e29T^{2} \)
47 \( 1 + 5.66e14iT - 1.25e30T^{2} \)
53 \( 1 - 3.33e15iT - 1.08e31T^{2} \)
59 \( 1 - 2.92e15T + 7.50e31T^{2} \)
61 \( 1 - 1.14e16iT - 1.36e32T^{2} \)
67 \( 1 - 2.86e16T + 7.40e32T^{2} \)
71 \( 1 + 5.39e16iT - 2.10e33T^{2} \)
73 \( 1 - 9.57e15T + 3.46e33T^{2} \)
79 \( 1 + 9.24e16iT - 1.43e34T^{2} \)
83 \( 1 + 3.30e17T + 3.49e34T^{2} \)
89 \( 1 + 2.93e17T + 1.22e35T^{2} \)
97 \( 1 + 8.93e17T + 5.77e35T^{2} \)
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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

−18.24365667294570791362286813375, −16.79896472064994700203127558662, −15.10703221038173426006407984062, −14.43039301623247228674464631264, −12.21838089855004343732935572858, −10.41809616143942876939303876518, −8.429544017435860675051194136318, −6.58063129620813053712878163714, −5.53677182518265762319668128170, −2.97642754870968107935540621096, 0.24747989445463186891062557437, 1.31875320724022738192591759146, 3.93031879356225416447939874594, 5.31857744730251433013087345494, 8.258544914032997236555638347341, 9.875559505702033270896334592962, 11.47316845640710055102276092295, 12.73384454798598253456769483999, 13.99671856800188009615260865781, 16.81396933987959420230372523552

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