Properties

Degree 2
Conductor $ 2^{3} $
Sign $0.344 - 0.938i$
Motivic weight 17
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (42.2 − 359. i)2-s + 1.60e4i·3-s + (−1.27e5 − 3.04e4i)4-s − 1.27e6i·5-s + (5.75e6 + 6.76e5i)6-s + 5.50e6·7-s + (−1.63e7 + 4.45e7i)8-s − 1.26e8·9-s + (−4.57e8 − 5.38e7i)10-s + 1.09e9i·11-s + (4.86e8 − 2.04e9i)12-s + 4.58e9i·13-s + (2.32e8 − 1.97e9i)14-s + 2.03e10·15-s + (1.53e10 + 7.75e9i)16-s − 3.61e10·17-s + ⋯
L(s)  = 1  + (0.116 − 0.993i)2-s + 1.40i·3-s + (−0.972 − 0.232i)4-s − 1.45i·5-s + (1.39 + 0.164i)6-s + 0.360·7-s + (−0.344 + 0.938i)8-s − 0.982·9-s + (−1.44 − 0.170i)10-s + 1.53i·11-s + (0.326 − 1.36i)12-s + 1.55i·13-s + (0.0421 − 0.358i)14-s + 2.05·15-s + (0.892 + 0.451i)16-s − 1.25·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8\)    =    \(2^{3}\)
\( \varepsilon \)  =  $0.344 - 0.938i$
motivic weight  =  \(17\)
character  :  $\chi_{8} (5, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 8,\ (\ :17/2),\ 0.344 - 0.938i)\)
\(L(9)\)  \(\approx\)  \(0.985823 + 0.688661i\)
\(L(\frac12)\)  \(\approx\)  \(0.985823 + 0.688661i\)
\(L(\frac{19}{2})\)   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 \( 1 + (-42.2 + 359. i)T \)
good3 \( 1 - 1.60e4iT - 1.29e8T^{2} \)
5 \( 1 + 1.27e6iT - 7.62e11T^{2} \)
7 \( 1 - 5.50e6T + 2.32e14T^{2} \)
11 \( 1 - 1.09e9iT - 5.05e17T^{2} \)
13 \( 1 - 4.58e9iT - 8.65e18T^{2} \)
17 \( 1 + 3.61e10T + 8.27e20T^{2} \)
19 \( 1 - 2.76e10iT - 5.48e21T^{2} \)
23 \( 1 - 2.20e11T + 1.41e23T^{2} \)
29 \( 1 - 2.89e11iT - 7.25e24T^{2} \)
31 \( 1 + 6.06e10T + 2.25e25T^{2} \)
37 \( 1 + 1.88e12iT - 4.56e26T^{2} \)
41 \( 1 - 6.37e13T + 2.61e27T^{2} \)
43 \( 1 - 1.04e14iT - 5.87e27T^{2} \)
47 \( 1 + 1.48e14T + 2.66e28T^{2} \)
53 \( 1 + 4.05e14iT - 2.05e29T^{2} \)
59 \( 1 + 7.77e14iT - 1.27e30T^{2} \)
61 \( 1 - 2.64e15iT - 2.24e30T^{2} \)
67 \( 1 + 1.78e15iT - 1.10e31T^{2} \)
71 \( 1 + 9.48e14T + 2.96e31T^{2} \)
73 \( 1 + 6.41e14T + 4.74e31T^{2} \)
79 \( 1 + 1.35e16T + 1.81e32T^{2} \)
83 \( 1 + 2.30e16iT - 4.21e32T^{2} \)
89 \( 1 + 1.39e16T + 1.37e33T^{2} \)
97 \( 1 + 1.33e15T + 5.95e33T^{2} \)
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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

−17.58270790011511649313766285887, −16.28836134546035218736590087833, −14.70749643852731432518499182991, −12.85501751803673698805186242360, −11.40988501932194158654932868275, −9.704071188380597678175570680620, −8.911198437101589664009615595623, −4.73600510902887622689286181885, −4.36276489275796661068400822358, −1.72892071410434151983222169508, 0.48336059363789582544197741461, 2.96628385141698808331269133597, 5.95859276083422139846295176815, 7.06523153772023132416238980497, 8.266023592159483223870702776426, 10.99739379389210167371297613035, 13.07534992316185426489809736652, 14.07235441343042257586570642897, 15.41899205717870236530049920895, 17.55649948328495256952131006071

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