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Properties

Label 2-2e3-8.3-c12-0-10
Degree $2$
Conductor $8$
Sign $-0.891 + 0.453i$
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  + (22.9 − 59.7i)2-s + 271.·3-s + (−3.04e3 − 2.74e3i)4-s − 1.10e4i·5-s + (6.22e3 − 1.62e4i)6-s − 5.87e4i·7-s + (−2.33e5 + 1.18e5i)8-s − 4.57e5·9-s + (−6.59e5 − 2.53e5i)10-s − 1.46e5·11-s + (−8.25e5 − 7.43e5i)12-s − 4.26e6i·13-s + (−3.51e6 − 1.34e6i)14-s − 2.99e6i·15-s + (1.74e6 + 1.66e7i)16-s + 2.95e7·17-s + ⋯
L(s)  = 1  + (0.358 − 0.933i)2-s + 0.372·3-s + (−0.743 − 0.669i)4-s − 0.706i·5-s + (0.133 − 0.347i)6-s − 0.499i·7-s + (−0.891 + 0.453i)8-s − 0.861·9-s + (−0.659 − 0.253i)10-s − 0.0828·11-s + (−0.276 − 0.248i)12-s − 0.883i·13-s + (−0.466 − 0.179i)14-s − 0.262i·15-s + (0.104 + 0.994i)16-s + 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.891 + 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.891 + 0.453i$
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: \((2,\ 8,\ (\ :6),\ -0.891 + 0.453i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.400111 - 1.66722i\)
\(L(\frac12)\) \(\approx\) \(0.400111 - 1.66722i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-22.9 + 59.7i)T \)
good3 \( 1 - 271.T + 5.31e5T^{2} \)
5 \( 1 + 1.10e4iT - 2.44e8T^{2} \)
7 \( 1 + 5.87e4iT - 1.38e10T^{2} \)
11 \( 1 + 1.46e5T + 3.13e12T^{2} \)
13 \( 1 + 4.26e6iT - 2.32e13T^{2} \)
17 \( 1 - 2.95e7T + 5.82e14T^{2} \)
19 \( 1 - 6.17e7T + 2.21e15T^{2} \)
23 \( 1 + 2.07e8iT - 2.19e16T^{2} \)
29 \( 1 - 4.01e8iT - 3.53e17T^{2} \)
31 \( 1 + 4.64e8iT - 7.87e17T^{2} \)
37 \( 1 - 4.99e9iT - 6.58e18T^{2} \)
41 \( 1 - 7.68e7T + 2.25e19T^{2} \)
43 \( 1 + 7.15e9T + 3.99e19T^{2} \)
47 \( 1 + 2.06e10iT - 1.16e20T^{2} \)
53 \( 1 - 5.81e9iT - 4.91e20T^{2} \)
59 \( 1 - 3.59e10T + 1.77e21T^{2} \)
61 \( 1 + 3.83e10iT - 2.65e21T^{2} \)
67 \( 1 + 8.74e10T + 8.18e21T^{2} \)
71 \( 1 - 1.98e11iT - 1.64e22T^{2} \)
73 \( 1 + 1.90e11T + 2.29e22T^{2} \)
79 \( 1 - 4.39e10iT - 5.90e22T^{2} \)
83 \( 1 - 3.26e11T + 1.06e23T^{2} \)
89 \( 1 - 1.81e11T + 2.46e23T^{2} \)
97 \( 1 - 4.70e10T + 6.93e23T^{2} \)
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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

−18.51226761648103060390443863281, −16.82256034353876947858304951416, −14.67702444147438148226544920916, −13.38411452774761428016050164759, −11.95074431040729603275026983553, −10.15888595799030871303686553594, −8.448316328993750450468386947552, −5.24253086285196935427044206751, −3.15476334098191105333761340239, −0.835750787407317372910257592154, 3.20554758924775429525750760784, 5.67421478804982519747588140235, 7.52785485242843908576856775145, 9.253558459930836578931865825854, 11.82896410859386901834000307667, 13.86506130258016828834278736251, 14.75976509478897662883754220088, 16.24743986347079137791664875605, 17.82117523608616419483774449627, 19.14086984386245115943064786748

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