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Properties

Label 2-2e3-8.5-c11-0-0
Degree $2$
Conductor $8$
Sign $-0.922 + 0.386i$
Analytic cond. $6.14674$
Root an. cond. $2.47926$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−44.8 + 5.96i)2-s + 614. i·3-s + (1.97e3 − 534. i)4-s + 4.39e3i·5-s + (−3.66e3 − 2.75e4i)6-s − 4.15e4·7-s + (−8.54e4 + 3.57e4i)8-s − 2.00e5·9-s + (−2.62e4 − 1.97e5i)10-s − 9.65e5i·11-s + (3.28e5 + 1.21e6i)12-s + 1.37e6i·13-s + (1.86e6 − 2.47e5i)14-s − 2.70e6·15-s + (3.62e6 − 2.11e6i)16-s − 3.12e6·17-s + ⋯
L(s)  = 1  + (−0.991 + 0.131i)2-s + 1.46i·3-s + (0.965 − 0.261i)4-s + 0.629i·5-s + (−0.192 − 1.44i)6-s − 0.934·7-s + (−0.922 + 0.386i)8-s − 1.13·9-s + (−0.0829 − 0.623i)10-s − 1.80i·11-s + (0.381 + 1.40i)12-s + 1.02i·13-s + (0.926 − 0.123i)14-s − 0.919·15-s + (0.863 − 0.504i)16-s − 0.533·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.922 + 0.386i$
Analytic conductor: \(6.14674\)
Root analytic conductor: \(2.47926\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :11/2),\ -0.922 + 0.386i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0925485 - 0.460853i\)
\(L(\frac12)\) \(\approx\) \(0.0925485 - 0.460853i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (44.8 - 5.96i)T \)
good3 \( 1 - 614. iT - 1.77e5T^{2} \)
5 \( 1 - 4.39e3iT - 4.88e7T^{2} \)
7 \( 1 + 4.15e4T + 1.97e9T^{2} \)
11 \( 1 + 9.65e5iT - 2.85e11T^{2} \)
13 \( 1 - 1.37e6iT - 1.79e12T^{2} \)
17 \( 1 + 3.12e6T + 3.42e13T^{2} \)
19 \( 1 - 4.95e6iT - 1.16e14T^{2} \)
23 \( 1 + 5.16e7T + 9.52e14T^{2} \)
29 \( 1 - 1.45e8iT - 1.22e16T^{2} \)
31 \( 1 - 3.49e7T + 2.54e16T^{2} \)
37 \( 1 - 2.21e7iT - 1.77e17T^{2} \)
41 \( 1 - 9.44e6T + 5.50e17T^{2} \)
43 \( 1 - 7.85e8iT - 9.29e17T^{2} \)
47 \( 1 + 1.72e9T + 2.47e18T^{2} \)
53 \( 1 - 7.35e8iT - 9.26e18T^{2} \)
59 \( 1 - 5.74e7iT - 3.01e19T^{2} \)
61 \( 1 - 7.93e9iT - 4.35e19T^{2} \)
67 \( 1 - 5.62e9iT - 1.22e20T^{2} \)
71 \( 1 - 1.32e10T + 2.31e20T^{2} \)
73 \( 1 - 5.44e9T + 3.13e20T^{2} \)
79 \( 1 + 1.46e9T + 7.47e20T^{2} \)
83 \( 1 + 4.76e10iT - 1.28e21T^{2} \)
89 \( 1 - 1.31e10T + 2.77e21T^{2} \)
97 \( 1 - 1.22e11T + 7.15e21T^{2} \)
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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

−19.73472262031706129555857926236, −18.54386865002120898385231345025, −16.44194956733978063768099382884, −16.06500622920321272762888555284, −14.39467277161726342183699210527, −11.26357652830512127741977500035, −10.13462428106284561508407477422, −8.836509898555188916515428294442, −6.26827156463139215601000983198, −3.31471826879444361215202352707, 0.34951559700265728868624102688, 2.10661802300179553854808186518, 6.59595168681665933174541748157, 7.937595883974495790852405471514, 9.806518087628529623560264105396, 12.19527703048493265868962402171, 12.96818688246268051807155397589, 15.59343574515804217552386896333, 17.35230070926115320732153997418, 18.18492203281307348079681056179

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