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Properties

Label 2-2e3-8.3-c18-0-0
Degree $2$
Conductor $8$
Sign $-0.683 - 0.729i$
Analytic cond. $16.4308$
Root an. cond. $4.05350$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−365. + 358. i)2-s − 3.34e4·3-s + (5.74e3 − 2.62e5i)4-s + 4.36e5i·5-s + (1.22e7 − 1.19e7i)6-s − 5.56e7i·7-s + (9.17e7 + 9.79e7i)8-s + 7.33e8·9-s + (−1.56e8 − 1.59e8i)10-s − 1.73e9·11-s + (−1.92e8 + 8.77e9i)12-s − 1.57e10i·13-s + (1.99e10 + 2.03e10i)14-s − 1.46e10i·15-s + (−6.86e10 − 3.01e9i)16-s − 4.47e10·17-s + ⋯
L(s)  = 1  + (−0.714 + 0.699i)2-s − 1.70·3-s + (0.0219 − 0.999i)4-s + 0.223i·5-s + (1.21 − 1.18i)6-s − 1.37i·7-s + (0.683 + 0.729i)8-s + 1.89·9-s + (−0.156 − 0.159i)10-s − 0.736·11-s + (−0.0372 + 1.70i)12-s − 1.48i·13-s + (0.964 + 0.986i)14-s − 0.379i·15-s + (−0.999 − 0.0438i)16-s − 0.377·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.683 - 0.729i$
Analytic conductor: \(16.4308\)
Root analytic conductor: \(4.05350\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :9),\ -0.683 - 0.729i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.0493257 + 0.113757i\)
\(L(\frac12)\) \(\approx\) \(0.0493257 + 0.113757i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (365. - 358. i)T \)
good3 \( 1 + 3.34e4T + 3.87e8T^{2} \)
5 \( 1 - 4.36e5iT - 3.81e12T^{2} \)
7 \( 1 + 5.56e7iT - 1.62e15T^{2} \)
11 \( 1 + 1.73e9T + 5.55e18T^{2} \)
13 \( 1 + 1.57e10iT - 1.12e20T^{2} \)
17 \( 1 + 4.47e10T + 1.40e22T^{2} \)
19 \( 1 + 3.79e11T + 1.04e23T^{2} \)
23 \( 1 - 7.33e11iT - 3.24e24T^{2} \)
29 \( 1 - 1.53e13iT - 2.10e26T^{2} \)
31 \( 1 + 4.57e13iT - 6.99e26T^{2} \)
37 \( 1 - 1.33e14iT - 1.68e28T^{2} \)
41 \( 1 + 2.64e14T + 1.07e29T^{2} \)
43 \( 1 - 6.42e13T + 2.52e29T^{2} \)
47 \( 1 - 9.41e14iT - 1.25e30T^{2} \)
53 \( 1 - 1.45e15iT - 1.08e31T^{2} \)
59 \( 1 + 1.62e16T + 7.50e31T^{2} \)
61 \( 1 - 7.69e15iT - 1.36e32T^{2} \)
67 \( 1 - 8.72e15T + 7.40e32T^{2} \)
71 \( 1 + 2.36e16iT - 2.10e33T^{2} \)
73 \( 1 + 4.05e15T + 3.46e33T^{2} \)
79 \( 1 - 9.28e16iT - 1.43e34T^{2} \)
83 \( 1 - 2.07e17T + 3.49e34T^{2} \)
89 \( 1 - 4.17e17T + 1.22e35T^{2} \)
97 \( 1 - 9.03e17T + 5.77e35T^{2} \)
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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

−17.41547981529885222531694121685, −16.70754337291306010016768157861, −15.31707415697919069697391547368, −13.11701050682788255912388028732, −10.86296671168182947964411983045, −10.36476513929482025754632783299, −7.59737723338860503208937745133, −6.29415631063629076872704851666, −4.85919610762294221491030544778, −0.887785095949743066607537950934, 0.10509085306912085040233711110, 1.98818494430013970341447732400, 4.77382153686199793288338865099, 6.53785051950549177285281764115, 8.881460067450103980833335997848, 10.63433065079625598005953235108, 11.80640250980076728551867872844, 12.66644229972213218335065835250, 15.85298515870686521547017207347, 16.90619431120121518594621119261

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