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Properties

Label 2-2e3-8.3-c12-0-6
Degree $2$
Conductor $8$
Sign $-0.839 + 0.543i$
Analytic cond. $7.31195$
Root an. cond. $2.70406$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−62.8 + 12.1i)2-s + 24.0·3-s + (3.79e3 − 1.53e3i)4-s + 2.51e4i·5-s + (−1.51e3 + 293. i)6-s − 1.90e5i·7-s + (−2.19e5 + 1.42e5i)8-s − 5.30e5·9-s + (−3.06e5 − 1.57e6i)10-s − 1.60e6·11-s + (9.15e4 − 3.69e4i)12-s − 3.13e6i·13-s + (2.32e6 + 1.19e7i)14-s + 6.05e5i·15-s + (1.20e7 − 1.16e7i)16-s − 1.87e7·17-s + ⋯
L(s)  = 1  + (−0.981 + 0.190i)2-s + 0.0330·3-s + (0.927 − 0.374i)4-s + 1.60i·5-s + (−0.0324 + 0.00629i)6-s − 1.61i·7-s + (−0.839 + 0.543i)8-s − 0.998·9-s + (−0.306 − 1.57i)10-s − 0.905·11-s + (0.0306 − 0.0123i)12-s − 0.648i·13-s + (0.308 + 1.58i)14-s + 0.0531i·15-s + (0.720 − 0.693i)16-s − 0.775·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.839 + 0.543i$
Analytic conductor: \(7.31195\)
Root analytic conductor: \(2.70406\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :6),\ -0.839 + 0.543i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.0251726 - 0.0851051i\)
\(L(\frac12)\) \(\approx\) \(0.0251726 - 0.0851051i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (62.8 - 12.1i)T \)
good3 \( 1 - 24.0T + 5.31e5T^{2} \)
5 \( 1 - 2.51e4iT - 2.44e8T^{2} \)
7 \( 1 + 1.90e5iT - 1.38e10T^{2} \)
11 \( 1 + 1.60e6T + 3.13e12T^{2} \)
13 \( 1 + 3.13e6iT - 2.32e13T^{2} \)
17 \( 1 + 1.87e7T + 5.82e14T^{2} \)
19 \( 1 + 2.37e7T + 2.21e15T^{2} \)
23 \( 1 + 9.22e7iT - 2.19e16T^{2} \)
29 \( 1 + 4.84e8iT - 3.53e17T^{2} \)
31 \( 1 - 6.86e8iT - 7.87e17T^{2} \)
37 \( 1 - 2.95e9iT - 6.58e18T^{2} \)
41 \( 1 + 4.28e9T + 2.25e19T^{2} \)
43 \( 1 - 5.06e9T + 3.99e19T^{2} \)
47 \( 1 - 8.78e9iT - 1.16e20T^{2} \)
53 \( 1 - 1.91e10iT - 4.91e20T^{2} \)
59 \( 1 + 3.49e9T + 1.77e21T^{2} \)
61 \( 1 + 2.44e10iT - 2.65e21T^{2} \)
67 \( 1 + 8.26e10T + 8.18e21T^{2} \)
71 \( 1 + 1.93e11iT - 1.64e22T^{2} \)
73 \( 1 - 1.67e11T + 2.29e22T^{2} \)
79 \( 1 - 1.86e11iT - 5.90e22T^{2} \)
83 \( 1 + 1.03e11T + 1.06e23T^{2} \)
89 \( 1 + 4.34e11T + 2.46e23T^{2} \)
97 \( 1 + 1.22e12T + 6.93e23T^{2} \)
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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

−18.09428997075356165114368207169, −17.06920720347351356095386884272, −15.27359225609956785069155191073, −13.95549798231904081871491024521, −10.98073810496660874304451648889, −10.36911526924374213354463826383, −7.87216355606217133590736004006, −6.55563985544091262671822436038, −2.83998557278829277960114792903, −0.05902993498742956098510830027, 2.17469550128057165514208641760, 5.55937071153567886442405128528, 8.451258783058421734065199773098, 9.133240715008434229430345272220, 11.56780906002823916373522674689, 12.74099977287831299248143598283, 15.45496735505224546052527536768, 16.58618578793424064389593510234, 17.92441798223113579978342537941, 19.37982520501107267679806055738

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