Properties

Degree 2
Conductor $ 2^{3} $
Sign $-0.692 + 0.721i$
Motivic weight 17
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−257. − 254. i)2-s − 4.83e3i·3-s + (1.83e3 + 1.31e5i)4-s − 5.24e5i·5-s + (−1.22e6 + 1.24e6i)6-s + 1.57e7·7-s + (3.28e7 − 3.42e7i)8-s + 1.05e8·9-s + (−1.33e8 + 1.35e8i)10-s + 2.44e8i·11-s + (6.33e8 − 8.89e6i)12-s − 2.66e9i·13-s + (−4.06e9 − 4.00e9i)14-s − 2.53e9·15-s + (−1.71e10 + 4.82e8i)16-s − 2.82e10·17-s + ⋯
L(s)  = 1  + (−0.712 − 0.702i)2-s − 0.425i·3-s + (0.0140 + 0.999i)4-s − 0.600i·5-s + (−0.298 + 0.302i)6-s + 1.03·7-s + (0.692 − 0.721i)8-s + 0.819·9-s + (−0.421 + 0.427i)10-s + 0.344i·11-s + (0.425 − 0.00597i)12-s − 0.905i·13-s + (−0.735 − 0.725i)14-s − 0.255·15-s + (−0.999 + 0.0280i)16-s − 0.983·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $-0.692 + 0.721i$
motivic weight  =  \(17\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8,\ (\ :17/2),\ -0.692 + 0.721i)\)
\(L(9)\)  \(\approx\)  \(0.509259 - 1.19376i\)
\(L(\frac12)\)  \(\approx\)  \(0.509259 - 1.19376i\)
\(L(\frac{19}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 2$,\(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (257. + 254. i)T \)
good3 \( 1 + 4.83e3iT - 1.29e8T^{2} \)
5 \( 1 + 5.24e5iT - 7.62e11T^{2} \)
7 \( 1 - 1.57e7T + 2.32e14T^{2} \)
11 \( 1 - 2.44e8iT - 5.05e17T^{2} \)
13 \( 1 + 2.66e9iT - 8.65e18T^{2} \)
17 \( 1 + 2.82e10T + 8.27e20T^{2} \)
19 \( 1 + 7.79e10iT - 5.48e21T^{2} \)
23 \( 1 + 3.66e11T + 1.41e23T^{2} \)
29 \( 1 - 2.78e11iT - 7.25e24T^{2} \)
31 \( 1 - 3.68e12T + 2.25e25T^{2} \)
37 \( 1 + 3.50e13iT - 4.56e26T^{2} \)
41 \( 1 + 3.67e13T + 2.61e27T^{2} \)
43 \( 1 + 1.25e14iT - 5.87e27T^{2} \)
47 \( 1 - 1.02e14T + 2.66e28T^{2} \)
53 \( 1 - 4.67e14iT - 2.05e29T^{2} \)
59 \( 1 - 4.08e13iT - 1.27e30T^{2} \)
61 \( 1 - 2.55e15iT - 2.24e30T^{2} \)
67 \( 1 + 2.46e15iT - 1.10e31T^{2} \)
71 \( 1 - 1.09e15T + 2.96e31T^{2} \)
73 \( 1 + 1.21e16T + 4.74e31T^{2} \)
79 \( 1 - 2.49e16T + 1.81e32T^{2} \)
83 \( 1 + 3.68e16iT - 4.21e32T^{2} \)
89 \( 1 + 4.15e16T + 1.37e33T^{2} \)
97 \( 1 - 7.13e15T + 5.95e33T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.44210843016758760201825051990, −15.65222640647288739353614054445, −13.32873622387385118068800007655, −12.13282122580669036074339836797, −10.56361089229579259669684455571, −8.770471221371494046032967043329, −7.39614835167933185159193148691, −4.50754750346270463133513197676, −2.06939227516634202756925143554, −0.71467164609970527284521361178, 1.63999979014838998478265782047, 4.57925033453795506103620844530, 6.60658437872296856327630208852, 8.224533239440868793513630980173, 9.950519099825975659597453863265, 11.26525425847194885656783100357, 14.03959583915538396705101094963, 15.17837605235193254000600980587, 16.47716530430271911043348914611, 17.98190683882842427828968799267

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