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Properties

Label 2-2e3-8.5-c11-0-1
Degree $2$
Conductor $8$
Sign $-0.907 - 0.421i$
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  + (16.7 + 42.0i)2-s − 284. i·3-s + (−1.48e3 + 1.40e3i)4-s + 1.04e4i·5-s + (1.19e4 − 4.75e3i)6-s − 7.05e4·7-s + (−8.40e4 − 3.90e4i)8-s + 9.62e4·9-s + (−4.40e5 + 1.75e5i)10-s + 1.77e5i·11-s + (4.00e5 + 4.23e5i)12-s + 1.12e6i·13-s + (−1.17e6 − 2.96e6i)14-s + 2.97e6·15-s + (2.34e5 − 4.18e6i)16-s + 6.79e6·17-s + ⋯
L(s)  = 1  + (0.369 + 0.929i)2-s − 0.675i·3-s + (−0.726 + 0.687i)4-s + 1.49i·5-s + (0.627 − 0.249i)6-s − 1.58·7-s + (−0.907 − 0.421i)8-s + 0.543·9-s + (−1.39 + 0.553i)10-s + 0.332i·11-s + (0.464 + 0.490i)12-s + 0.837i·13-s + (−0.586 − 1.47i)14-s + 1.01·15-s + (0.0558 − 0.998i)16-s + 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.421i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.907 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.907 - 0.421i$
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: \((2,\ 8,\ (\ :11/2),\ -0.907 - 0.421i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.271498 + 1.22964i\)
\(L(\frac12)\) \(\approx\) \(0.271498 + 1.22964i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-16.7 - 42.0i)T \)
good3 \( 1 + 284. iT - 1.77e5T^{2} \)
5 \( 1 - 1.04e4iT - 4.88e7T^{2} \)
7 \( 1 + 7.05e4T + 1.97e9T^{2} \)
11 \( 1 - 1.77e5iT - 2.85e11T^{2} \)
13 \( 1 - 1.12e6iT - 1.79e12T^{2} \)
17 \( 1 - 6.79e6T + 3.42e13T^{2} \)
19 \( 1 - 3.43e6iT - 1.16e14T^{2} \)
23 \( 1 - 2.44e7T + 9.52e14T^{2} \)
29 \( 1 - 6.35e7iT - 1.22e16T^{2} \)
31 \( 1 + 8.31e7T + 2.54e16T^{2} \)
37 \( 1 - 4.05e8iT - 1.77e17T^{2} \)
41 \( 1 + 1.21e9T + 5.50e17T^{2} \)
43 \( 1 - 8.44e8iT - 9.29e17T^{2} \)
47 \( 1 - 1.31e9T + 2.47e18T^{2} \)
53 \( 1 - 1.71e8iT - 9.26e18T^{2} \)
59 \( 1 - 8.98e9iT - 3.01e19T^{2} \)
61 \( 1 + 1.06e10iT - 4.35e19T^{2} \)
67 \( 1 + 4.61e9iT - 1.22e20T^{2} \)
71 \( 1 + 3.76e9T + 2.31e20T^{2} \)
73 \( 1 - 2.35e10T + 3.13e20T^{2} \)
79 \( 1 - 1.77e10T + 7.47e20T^{2} \)
83 \( 1 - 4.42e9iT - 1.28e21T^{2} \)
89 \( 1 + 7.93e9T + 2.77e21T^{2} \)
97 \( 1 - 1.20e10T + 7.15e21T^{2} \)
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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

−19.00562157236335079373002535540, −18.46259682908174825833869139231, −16.58787039387542077087346545192, −15.14784482446346827086889698675, −13.75762640211590981064792256692, −12.41782081913419460461331691261, −9.842837416026770909404381791105, −7.21241286924568727115504884357, −6.45407461363685241699403304981, −3.30161979987995016174650526028, 0.66638678585317323938868056002, 3.58760854203389509008945204736, 5.33553346932671623349764545337, 9.111987757024184618620522311156, 10.16666834312276241394043617364, 12.44075907606665968342317845396, 13.22180435922837456700697883898, 15.55825372573406298159561019931, 16.74877887621199266307368690451, 18.97428549373063496456043514980

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