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Properties

Label 2-2e3-8.5-c19-0-4
Degree $2$
Conductor $8$
Sign $0.844 + 0.534i$
Analytic cond. $18.3053$
Root an. cond. $4.27847$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−238. − 683. i)2-s + 2.08e3i·3-s + (−4.10e5 + 3.25e5i)4-s − 1.39e6i·5-s + (1.42e6 − 4.96e5i)6-s − 1.59e8·7-s + (3.20e8 + 2.03e8i)8-s + 1.15e9·9-s + (−9.50e8 + 3.31e8i)10-s + 1.34e10i·11-s + (−6.78e8 − 8.55e8i)12-s − 2.91e10i·13-s + (3.80e10 + 1.09e11i)14-s + 2.89e9·15-s + (6.23e10 − 2.67e11i)16-s + 3.85e11·17-s + ⋯
L(s)  = 1  + (−0.329 − 0.944i)2-s + 0.0610i·3-s + (−0.783 + 0.621i)4-s − 0.318i·5-s + (0.0576 − 0.0201i)6-s − 1.49·7-s + (0.844 + 0.534i)8-s + 0.996·9-s + (−0.300 + 0.104i)10-s + 1.71i·11-s + (−0.0379 − 0.0478i)12-s − 0.763i·13-s + (0.491 + 1.41i)14-s + 0.0194·15-s + (0.226 − 0.973i)16-s + 0.787·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.844 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.844 + 0.534i$
Analytic conductor: \(18.3053\)
Root analytic conductor: \(4.27847\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :19/2),\ 0.844 + 0.534i)\)

Particular Values

\(L(10)\) \(\approx\) \(1.20824 - 0.350303i\)
\(L(\frac12)\) \(\approx\) \(1.20824 - 0.350303i\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (238. + 683. i)T \)
good3 \( 1 - 2.08e3iT - 1.16e9T^{2} \)
5 \( 1 + 1.39e6iT - 1.90e13T^{2} \)
7 \( 1 + 1.59e8T + 1.13e16T^{2} \)
11 \( 1 - 1.34e10iT - 6.11e19T^{2} \)
13 \( 1 + 2.91e10iT - 1.46e21T^{2} \)
17 \( 1 - 3.85e11T + 2.39e23T^{2} \)
19 \( 1 + 1.37e12iT - 1.97e24T^{2} \)
23 \( 1 - 7.61e12T + 7.46e25T^{2} \)
29 \( 1 + 2.78e13iT - 6.10e27T^{2} \)
31 \( 1 - 3.08e13T + 2.16e28T^{2} \)
37 \( 1 - 1.10e15iT - 6.24e29T^{2} \)
41 \( 1 - 1.56e15T + 4.39e30T^{2} \)
43 \( 1 - 3.24e15iT - 1.08e31T^{2} \)
47 \( 1 - 1.13e16T + 5.88e31T^{2} \)
53 \( 1 - 2.40e16iT - 5.77e32T^{2} \)
59 \( 1 - 6.95e16iT - 4.42e33T^{2} \)
61 \( 1 + 1.01e17iT - 8.34e33T^{2} \)
67 \( 1 + 3.30e16iT - 4.95e34T^{2} \)
71 \( 1 - 1.12e17T + 1.49e35T^{2} \)
73 \( 1 + 2.72e17T + 2.53e35T^{2} \)
79 \( 1 + 1.25e18T + 1.13e36T^{2} \)
83 \( 1 - 2.08e18iT - 2.90e36T^{2} \)
89 \( 1 + 2.36e18T + 1.09e37T^{2} \)
97 \( 1 - 1.41e19T + 5.60e37T^{2} \)
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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

−17.03958043213535624016954586873, −15.46769730394632517987372629269, −13.04373581734241343267793354351, −12.46952115251911054111361432866, −10.24259201536176484517912341841, −9.399188353923566930246674331795, −7.22350061664427845966606986449, −4.57165930986141413681976138035, −2.87620629719959687081800520225, −0.969806628548077081821495839664, 0.75632830801987315545711665525, 3.60693058158510850229609463849, 5.92600048619061112449495286709, 7.11495053821642004573583648706, 9.017327658207126030700462937687, 10.38817642015107376564844916836, 12.89627283802762096077380635845, 14.21564449504577477763757467869, 15.98834900119197983658033413504, 16.60383875677295266308501150485

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