Properties

Degree $2$
Conductor $8$
Sign $0.999 - 0.0259i$
Motivic weight $17$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−362. + 3.13i)2-s − 1.38e4i·3-s + (1.31e5 − 2.27e3i)4-s + 6.65e5i·5-s + (4.34e4 + 5.02e6i)6-s − 7.57e6·7-s + (−4.74e7 + 1.23e6i)8-s − 6.31e7·9-s + (−2.08e6 − 2.40e8i)10-s + 1.97e8i·11-s + (−3.14e7 − 1.81e9i)12-s + 4.74e9i·13-s + (2.74e9 − 2.37e7i)14-s + 9.22e9·15-s + (1.71e10 − 5.95e8i)16-s + 3.37e10·17-s + ⋯
L(s)  = 1  + (−0.999 + 0.00866i)2-s − 1.22i·3-s + (0.999 − 0.0173i)4-s + 0.761i·5-s + (0.0105 + 1.22i)6-s − 0.496·7-s + (−0.999 + 0.0259i)8-s − 0.489·9-s + (−0.00659 − 0.761i)10-s + 0.278i·11-s + (−0.0211 − 1.22i)12-s + 1.61i·13-s + (0.496 − 0.00430i)14-s + 0.929·15-s + (0.999 − 0.0346i)16-s + 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0259i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.999 - 0.0259i$
Motivic weight: \(17\)
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :17/2),\ 0.999 - 0.0259i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.08147 + 0.0140519i\)
\(L(\frac12)\) \(\approx\) \(1.08147 + 0.0140519i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (362. - 3.13i)T \)
good3 \( 1 + 1.38e4iT - 1.29e8T^{2} \)
5 \( 1 - 6.65e5iT - 7.62e11T^{2} \)
7 \( 1 + 7.57e6T + 2.32e14T^{2} \)
11 \( 1 - 1.97e8iT - 5.05e17T^{2} \)
13 \( 1 - 4.74e9iT - 8.65e18T^{2} \)
17 \( 1 - 3.37e10T + 8.27e20T^{2} \)
19 \( 1 + 1.00e11iT - 5.48e21T^{2} \)
23 \( 1 - 3.16e11T + 1.41e23T^{2} \)
29 \( 1 - 3.54e12iT - 7.25e24T^{2} \)
31 \( 1 - 2.76e11T + 2.25e25T^{2} \)
37 \( 1 + 2.12e13iT - 4.56e26T^{2} \)
41 \( 1 - 8.47e13T + 2.61e27T^{2} \)
43 \( 1 - 1.38e14iT - 5.87e27T^{2} \)
47 \( 1 - 8.21e13T + 2.66e28T^{2} \)
53 \( 1 - 3.16e14iT - 2.05e29T^{2} \)
59 \( 1 + 3.29e14iT - 1.27e30T^{2} \)
61 \( 1 - 6.60e14iT - 2.24e30T^{2} \)
67 \( 1 - 3.64e15iT - 1.10e31T^{2} \)
71 \( 1 - 9.19e15T + 2.96e31T^{2} \)
73 \( 1 + 5.44e15T + 4.74e31T^{2} \)
79 \( 1 - 1.06e16T + 1.81e32T^{2} \)
83 \( 1 - 9.38e15iT - 4.21e32T^{2} \)
89 \( 1 - 2.24e16T + 1.37e33T^{2} \)
97 \( 1 + 4.00e16T + 5.95e33T^{2} \)
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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

−17.84046014871900871541508805088, −16.32691227338549058333272217255, −14.45916172797475659995087708821, −12.59134893637959526193557021242, −11.13583490064774578893768389328, −9.297042333426768432410034465215, −7.32754030067625590200203667942, −6.58768709989258005296661228740, −2.65632942381159082257042719693, −1.12854597882151162986182962752, 0.76062552356690262625588702070, 3.34416092327149839940736315786, 5.56416140863468238443460693757, 8.087742344461292888657965693134, 9.590720423767112409130980224019, 10.53048149139877749683420452479, 12.48651191962340683687916381779, 15.10428909378665905900397419574, 16.19331095937044013838356671630, 17.07122107663213279601170210049

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