Properties

Degree $2$
Conductor $8$
Sign $-0.268 - 0.963i$
Motivic weight $17$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (328. + 151. i)2-s − 4.24e3i·3-s + (8.49e4 + 9.98e4i)4-s + 6.63e5i·5-s + (6.45e5 − 1.39e6i)6-s − 1.66e7·7-s + (1.27e7 + 4.57e7i)8-s + 1.11e8·9-s + (−1.00e8 + 2.18e8i)10-s + 1.05e9i·11-s + (4.24e8 − 3.60e8i)12-s − 9.19e7i·13-s + (−5.47e9 − 2.53e9i)14-s + 2.82e9·15-s + (−2.75e9 + 1.69e10i)16-s − 1.98e10·17-s + ⋯
L(s)  = 1  + (0.907 + 0.419i)2-s − 0.373i·3-s + (0.648 + 0.761i)4-s + 0.760i·5-s + (0.156 − 0.339i)6-s − 1.09·7-s + (0.268 + 0.963i)8-s + 0.860·9-s + (−0.318 + 0.690i)10-s + 1.48i·11-s + (0.284 − 0.242i)12-s − 0.0312i·13-s + (−0.992 − 0.458i)14-s + 0.284·15-s + (−0.160 + 0.987i)16-s − 0.689·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.268 - 0.963i$
Motivic weight: \(17\)
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :17/2),\ -0.268 - 0.963i)\)

Particular Values

\(L(9)\) \(\approx\) \(1.64211 + 2.16298i\)
\(L(\frac12)\) \(\approx\) \(1.64211 + 2.16298i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-328. - 151. i)T \)
good3 \( 1 + 4.24e3iT - 1.29e8T^{2} \)
5 \( 1 - 6.63e5iT - 7.62e11T^{2} \)
7 \( 1 + 1.66e7T + 2.32e14T^{2} \)
11 \( 1 - 1.05e9iT - 5.05e17T^{2} \)
13 \( 1 + 9.19e7iT - 8.65e18T^{2} \)
17 \( 1 + 1.98e10T + 8.27e20T^{2} \)
19 \( 1 - 8.44e10iT - 5.48e21T^{2} \)
23 \( 1 + 2.72e10T + 1.41e23T^{2} \)
29 \( 1 + 3.75e12iT - 7.25e24T^{2} \)
31 \( 1 - 5.36e12T + 2.25e25T^{2} \)
37 \( 1 + 1.92e13iT - 4.56e26T^{2} \)
41 \( 1 - 5.42e13T + 2.61e27T^{2} \)
43 \( 1 + 3.96e13iT - 5.87e27T^{2} \)
47 \( 1 - 4.48e13T + 2.66e28T^{2} \)
53 \( 1 + 7.81e14iT - 2.05e29T^{2} \)
59 \( 1 - 1.07e15iT - 1.27e30T^{2} \)
61 \( 1 - 1.92e15iT - 2.24e30T^{2} \)
67 \( 1 - 3.68e15iT - 1.10e31T^{2} \)
71 \( 1 + 1.02e16T + 2.96e31T^{2} \)
73 \( 1 - 1.25e16T + 4.74e31T^{2} \)
79 \( 1 - 3.24e15T + 1.81e32T^{2} \)
83 \( 1 + 4.28e15iT - 4.21e32T^{2} \)
89 \( 1 - 6.95e16T + 1.37e33T^{2} \)
97 \( 1 + 7.74e16T + 5.95e33T^{2} \)
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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

−17.76140554465044456079915534332, −16.00009749951639998544242524354, −14.88134595963994379378501156734, −13.26004488889856699467292581841, −12.22187575913595672390977010457, −10.11127309234066276949016446941, −7.37657687486616011084659147523, −6.38609495789129591513581979712, −4.11754593410667581352617618277, −2.33131756261166549602944838397, 0.842591543557848077890502433286, 3.15373025124775396266202590568, 4.72458911276472190758276880665, 6.47694610608886868152917513790, 9.249176559491100788260481351008, 10.85913758134279591519191290380, 12.64487619703354681005926113574, 13.59473240751302874472901323153, 15.62053849916215171617439854204, 16.39635780212676505325872844125

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