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Properties

Label 2-2e3-8.5-c13-0-5
Degree $2$
Conductor $8$
Sign $0.640 - 0.767i$
Analytic cond. $8.57847$
Root an. cond. $2.92890$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (65.8 + 62.0i)2-s − 1.23e3i·3-s + (489. + 8.17e3i)4-s + 2.52e4i·5-s + (7.64e4 − 8.11e4i)6-s + 6.08e5·7-s + (−4.75e5 + 5.69e5i)8-s + 7.79e4·9-s + (−1.56e6 + 1.66e6i)10-s + 7.12e6i·11-s + (1.00e7 − 6.02e5i)12-s − 1.12e7i·13-s + (4.00e7 + 3.77e7i)14-s + 3.11e7·15-s + (−6.66e7 + 8.00e6i)16-s − 5.09e7·17-s + ⋯
L(s)  = 1  + (0.727 + 0.685i)2-s − 0.975i·3-s + (0.0597 + 0.998i)4-s + 0.723i·5-s + (0.668 − 0.709i)6-s + 1.95·7-s + (−0.640 + 0.767i)8-s + 0.0488·9-s + (−0.495 + 0.526i)10-s + 1.21i·11-s + (0.973 − 0.0582i)12-s − 0.648i·13-s + (1.42 + 1.33i)14-s + 0.705·15-s + (−0.992 + 0.119i)16-s − 0.511·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.640 - 0.767i$
Analytic conductor: \(8.57847\)
Root analytic conductor: \(2.92890\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :13/2),\ 0.640 - 0.767i)\)

Particular Values

\(L(7)\) \(\approx\) \(2.55512 + 1.19527i\)
\(L(\frac12)\) \(\approx\) \(2.55512 + 1.19527i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-65.8 - 62.0i)T \)
good3 \( 1 + 1.23e3iT - 1.59e6T^{2} \)
5 \( 1 - 2.52e4iT - 1.22e9T^{2} \)
7 \( 1 - 6.08e5T + 9.68e10T^{2} \)
11 \( 1 - 7.12e6iT - 3.45e13T^{2} \)
13 \( 1 + 1.12e7iT - 3.02e14T^{2} \)
17 \( 1 + 5.09e7T + 9.90e15T^{2} \)
19 \( 1 + 1.86e8iT - 4.20e16T^{2} \)
23 \( 1 + 3.07e8T + 5.04e17T^{2} \)
29 \( 1 - 4.74e8iT - 1.02e19T^{2} \)
31 \( 1 + 2.41e9T + 2.44e19T^{2} \)
37 \( 1 + 1.55e10iT - 2.43e20T^{2} \)
41 \( 1 + 2.32e10T + 9.25e20T^{2} \)
43 \( 1 - 6.55e10iT - 1.71e21T^{2} \)
47 \( 1 + 6.84e10T + 5.46e21T^{2} \)
53 \( 1 + 9.64e10iT - 2.60e22T^{2} \)
59 \( 1 + 3.22e11iT - 1.04e23T^{2} \)
61 \( 1 + 6.66e11iT - 1.61e23T^{2} \)
67 \( 1 - 5.91e11iT - 5.48e23T^{2} \)
71 \( 1 - 8.43e11T + 1.16e24T^{2} \)
73 \( 1 + 7.21e11T + 1.67e24T^{2} \)
79 \( 1 - 5.78e11T + 4.66e24T^{2} \)
83 \( 1 + 1.54e11iT - 8.87e24T^{2} \)
89 \( 1 - 2.38e12T + 2.19e25T^{2} \)
97 \( 1 + 1.26e13T + 6.73e25T^{2} \)
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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

−18.01190991297890747425172424913, −17.71589917394948224495098286825, −15.22840407429498477697154768548, −14.29071813278238700296139301621, −12.76940851115830205490987300544, −11.28660286471362163711734934589, −7.977332520532549214505395441524, −6.95079489358404702701397500004, −4.79021458568723709447934504244, −2.05597772219470668917871605546, 1.47206363951694865860601046652, 4.13847733670970732735505770859, 5.23353068686638011330789676137, 8.727530566140184871235034888812, 10.63564263509288000676105637554, 11.78285288767715316408471987830, 13.81902721714609421196886197479, 15.02879127992900801465866422669, 16.55995466000176649178516657766, 18.52347794703156014354139906089

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