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Properties

Label 2-2e3-8.5-c11-0-4
Degree $2$
Conductor $8$
Sign $0.703 - 0.710i$
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  + (43.6 − 11.7i)2-s + 381. i·3-s + (1.77e3 − 1.02e3i)4-s + 8.93e3i·5-s + (4.49e3 + 1.66e4i)6-s + 1.01e4·7-s + (6.52e4 − 6.58e4i)8-s + 3.14e4·9-s + (1.05e5 + 3.90e5i)10-s + 4.08e5i·11-s + (3.92e5 + 6.75e5i)12-s − 1.94e6i·13-s + (4.44e5 − 1.19e5i)14-s − 3.41e6·15-s + (2.07e6 − 3.64e6i)16-s − 9.64e6·17-s + ⋯
L(s)  = 1  + (0.965 − 0.260i)2-s + 0.906i·3-s + (0.864 − 0.502i)4-s + 1.27i·5-s + (0.236 + 0.875i)6-s + 0.228·7-s + (0.703 − 0.710i)8-s + 0.177·9-s + (0.332 + 1.23i)10-s + 0.764i·11-s + (0.455 + 0.784i)12-s − 1.45i·13-s + (0.220 − 0.0595i)14-s − 1.16·15-s + (0.494 − 0.869i)16-s − 1.64·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.703 - 0.710i$
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: \((2,\ 8,\ (\ :11/2),\ 0.703 - 0.710i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.61072 + 1.08858i\)
\(L(\frac12)\) \(\approx\) \(2.61072 + 1.08858i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-43.6 + 11.7i)T \)
good3 \( 1 - 381. iT - 1.77e5T^{2} \)
5 \( 1 - 8.93e3iT - 4.88e7T^{2} \)
7 \( 1 - 1.01e4T + 1.97e9T^{2} \)
11 \( 1 - 4.08e5iT - 2.85e11T^{2} \)
13 \( 1 + 1.94e6iT - 1.79e12T^{2} \)
17 \( 1 + 9.64e6T + 3.42e13T^{2} \)
19 \( 1 + 1.27e7iT - 1.16e14T^{2} \)
23 \( 1 - 2.34e7T + 9.52e14T^{2} \)
29 \( 1 + 4.83e7iT - 1.22e16T^{2} \)
31 \( 1 - 2.63e8T + 2.54e16T^{2} \)
37 \( 1 + 7.91e7iT - 1.77e17T^{2} \)
41 \( 1 + 6.96e8T + 5.50e17T^{2} \)
43 \( 1 - 4.03e8iT - 9.29e17T^{2} \)
47 \( 1 + 5.59e8T + 2.47e18T^{2} \)
53 \( 1 + 2.88e9iT - 9.26e18T^{2} \)
59 \( 1 - 9.84e9iT - 3.01e19T^{2} \)
61 \( 1 - 5.65e9iT - 4.35e19T^{2} \)
67 \( 1 + 1.28e10iT - 1.22e20T^{2} \)
71 \( 1 + 3.63e9T + 2.31e20T^{2} \)
73 \( 1 + 2.56e9T + 3.13e20T^{2} \)
79 \( 1 + 2.59e10T + 7.47e20T^{2} \)
83 \( 1 - 3.80e10iT - 1.28e21T^{2} \)
89 \( 1 + 2.00e10T + 2.77e21T^{2} \)
97 \( 1 + 6.90e10T + 7.15e21T^{2} \)
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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

−19.68850943154420162671191045128, −17.83616242537870241206231794804, −15.41306039500285378322643951498, −15.09918747609134769198165378241, −13.24267866794415572573364078182, −11.14447369839344902170573410448, −10.16384391228315439071502499614, −6.87389485697997156822447908785, −4.65805586266763413730864706536, −2.82667285396352382805119582853, 1.61090823087201124509438746805, 4.56030057871729415232598019412, 6.58142650165103934752555845151, 8.454866409377651228912527579046, 11.65811429700907389279532161049, 12.88253334129408767221824512900, 13.88558989504150370775387479043, 15.93171635907400655452883979812, 17.10733797553272260186188507954, 19.08222033361765026277909248930

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