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Properties

Label 2-2e3-8.5-c5-0-1
Degree $2$
Conductor $8$
Sign $0.814 - 0.580i$
Analytic cond. $1.28307$
Root an. cond. $1.13272$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.77 + 4.21i)2-s − 3.25i·3-s + (−3.54 + 31.8i)4-s − 73.9i·5-s + (13.7 − 12.2i)6-s − 112.·7-s + (−147. + 105. i)8-s + 232.·9-s + (311. − 278. i)10-s + 575. i·11-s + (103. + 11.5i)12-s − 117. i·13-s + (−425. − 475. i)14-s − 240.·15-s + (−998. − 225. i)16-s − 223.·17-s + ⋯
L(s)  = 1  + (0.666 + 0.745i)2-s − 0.208i·3-s + (−0.110 + 0.993i)4-s − 1.32i·5-s + (0.155 − 0.139i)6-s − 0.869·7-s + (−0.814 + 0.580i)8-s + 0.956·9-s + (0.985 − 0.882i)10-s + 1.43i·11-s + (0.207 + 0.0231i)12-s − 0.193i·13-s + (−0.579 − 0.647i)14-s − 0.276·15-s + (−0.975 − 0.220i)16-s − 0.187·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.814 - 0.580i$
Analytic conductor: \(1.28307\)
Root analytic conductor: \(1.13272\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :5/2),\ 0.814 - 0.580i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.32687 + 0.424253i\)
\(L(\frac12)\) \(\approx\) \(1.32687 + 0.424253i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.77 - 4.21i)T \)
good3 \( 1 + 3.25iT - 243T^{2} \)
5 \( 1 + 73.9iT - 3.12e3T^{2} \)
7 \( 1 + 112.T + 1.68e4T^{2} \)
11 \( 1 - 575. iT - 1.61e5T^{2} \)
13 \( 1 + 117. iT - 3.71e5T^{2} \)
17 \( 1 + 223.T + 1.41e6T^{2} \)
19 \( 1 + 1.75e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.36e3T + 6.43e6T^{2} \)
29 \( 1 - 3.86e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.59e3T + 2.86e7T^{2} \)
37 \( 1 + 4.73e3iT - 6.93e7T^{2} \)
41 \( 1 - 8.15e3T + 1.15e8T^{2} \)
43 \( 1 + 4.92e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.10e4T + 2.29e8T^{2} \)
53 \( 1 - 1.27e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.41e4iT - 7.14e8T^{2} \)
61 \( 1 - 4.20e4iT - 8.44e8T^{2} \)
67 \( 1 + 5.41e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.38e4T + 1.80e9T^{2} \)
73 \( 1 + 3.12e4T + 2.07e9T^{2} \)
79 \( 1 + 5.02e4T + 3.07e9T^{2} \)
83 \( 1 - 4.37e4iT - 3.93e9T^{2} \)
89 \( 1 - 6.44e4T + 5.58e9T^{2} \)
97 \( 1 + 6.23e4T + 8.58e9T^{2} \)
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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

−21.23093957885846219775091287499, −19.94192191080030960654387549031, −17.77970872257964054622090957177, −16.41363436814263299422483944269, −15.32388784117557855982143589838, −13.07464035345282286936694309402, −12.55298805903256300934222531860, −9.233917740109243373165274818869, −7.10216428735290904435596455564, −4.71880582775102599709882316828, 3.37396394190873989545603734583, 6.41092439415696637848192265195, 9.921819300493475266502780784502, 11.16966677750468474291321996939, 13.11881435984734104265713163009, 14.52953575703698600552459463057, 15.98056003575441920927910929848, 18.64891309708910917676946608020, 19.17603479818966679272018001677, 21.17612413933424290806263970973

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