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Properties

Label 2-2e3-8.5-c21-0-10
Degree $2$
Conductor $8$
Sign $-0.121 - 0.992i$
Analytic cond. $22.3581$
Root an. cond. $4.72844$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.28e3 + 672. i)2-s + 6.03e4i·3-s + (1.19e6 + 1.72e6i)4-s − 1.19e7i·5-s + (−4.05e7 + 7.73e7i)6-s + 8.57e8·7-s + (3.69e8 + 3.01e9i)8-s + 6.82e9·9-s + (8.03e9 − 1.53e10i)10-s + 3.73e10i·11-s + (−1.04e11 + 7.19e10i)12-s + 4.68e11i·13-s + (1.09e12 + 5.76e11i)14-s + 7.20e11·15-s + (−1.55e12 + 4.11e12i)16-s − 7.52e12·17-s + ⋯
L(s)  = 1  + (0.885 + 0.464i)2-s + 0.589i·3-s + (0.568 + 0.822i)4-s − 0.546i·5-s + (−0.273 + 0.522i)6-s + 1.14·7-s + (0.121 + 0.992i)8-s + 0.652·9-s + (0.253 − 0.484i)10-s + 0.433i·11-s + (−0.485 + 0.335i)12-s + 0.942i·13-s + (1.01 + 0.532i)14-s + 0.322·15-s + (−0.353 + 0.935i)16-s − 0.905·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.121 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.121 - 0.992i$
Analytic conductor: \(22.3581\)
Root analytic conductor: \(4.72844\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :21/2),\ -0.121 - 0.992i)\)

Particular Values

\(L(11)\) \(\approx\) \(2.50251 + 2.82765i\)
\(L(\frac12)\) \(\approx\) \(2.50251 + 2.82765i\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28e3 - 672. i)T \)
good3 \( 1 - 6.03e4iT - 1.04e10T^{2} \)
5 \( 1 + 1.19e7iT - 4.76e14T^{2} \)
7 \( 1 - 8.57e8T + 5.58e17T^{2} \)
11 \( 1 - 3.73e10iT - 7.40e21T^{2} \)
13 \( 1 - 4.68e11iT - 2.47e23T^{2} \)
17 \( 1 + 7.52e12T + 6.90e25T^{2} \)
19 \( 1 + 1.69e13iT - 7.14e26T^{2} \)
23 \( 1 + 1.71e14T + 3.94e28T^{2} \)
29 \( 1 - 3.63e15iT - 5.13e30T^{2} \)
31 \( 1 - 1.78e15T + 2.08e31T^{2} \)
37 \( 1 + 1.66e16iT - 8.55e32T^{2} \)
41 \( 1 - 8.59e16T + 7.38e33T^{2} \)
43 \( 1 + 2.20e17iT - 2.00e34T^{2} \)
47 \( 1 - 3.64e17T + 1.30e35T^{2} \)
53 \( 1 + 1.49e18iT - 1.62e36T^{2} \)
59 \( 1 - 2.76e18iT - 1.54e37T^{2} \)
61 \( 1 + 3.78e18iT - 3.10e37T^{2} \)
67 \( 1 + 2.47e19iT - 2.22e38T^{2} \)
71 \( 1 + 1.31e19T + 7.52e38T^{2} \)
73 \( 1 - 1.62e19T + 1.34e39T^{2} \)
79 \( 1 + 7.88e19T + 7.08e39T^{2} \)
83 \( 1 - 1.83e20iT - 1.99e40T^{2} \)
89 \( 1 + 1.06e20T + 8.65e40T^{2} \)
97 \( 1 - 1.14e21T + 5.27e41T^{2} \)
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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

−16.46358989873643622792847826415, −15.33315461748192171908808092785, −14.07138098531536759237958302869, −12.49827899833447173233712332770, −11.02029788596951980379382139663, −8.817601337561381160963562369300, −7.08365683266143971966145696619, −4.98535187466030257138186777782, −4.20102866199878606011852251641, −1.87372401437026518150448432406, 1.08939655775878604980774193881, 2.46708807606223595285953605055, 4.35577295356355664745250856808, 6.11840820908660081934778720360, 7.74979924829918070505849733375, 10.37901915809672215493058273435, 11.63477575024164599784613924792, 13.08525155168354620504927248176, 14.32677185972219354429933845442, 15.57090182052240924198979665520

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