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Properties

Label 2-2e3-8.5-c21-0-12
Degree $2$
Conductor $8$
Sign $-0.819 + 0.573i$
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  + (−962. − 1.08e3i)2-s − 8.55e4i·3-s + (−2.43e5 + 2.08e6i)4-s + 2.31e7i·5-s + (−9.25e7 + 8.23e7i)6-s − 3.19e8·7-s + (2.48e9 − 1.74e9i)8-s + 3.13e9·9-s + (2.50e10 − 2.22e10i)10-s − 6.15e10i·11-s + (1.78e11 + 2.08e10i)12-s + 1.45e11i·13-s + (3.07e11 + 3.45e11i)14-s + 1.97e12·15-s + (−4.27e12 − 1.01e12i)16-s + 9.09e12·17-s + ⋯
L(s)  = 1  + (−0.664 − 0.747i)2-s − 0.836i·3-s + (−0.116 + 0.993i)4-s + 1.05i·5-s + (−0.625 + 0.556i)6-s − 0.427·7-s + (0.819 − 0.573i)8-s + 0.299·9-s + (0.791 − 0.704i)10-s − 0.715i·11-s + (0.831 + 0.0970i)12-s + 0.292i·13-s + (0.283 + 0.319i)14-s + 0.886·15-s + (−0.973 − 0.230i)16-s + 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.819 + 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $-0.819 + 0.573i$
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: \((2,\ 8,\ (\ :21/2),\ -0.819 + 0.573i)\)

Particular Values

\(L(11)\) \(\approx\) \(0.283697 - 0.899602i\)
\(L(\frac12)\) \(\approx\) \(0.283697 - 0.899602i\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (962. + 1.08e3i)T \)
good3 \( 1 + 8.55e4iT - 1.04e10T^{2} \)
5 \( 1 - 2.31e7iT - 4.76e14T^{2} \)
7 \( 1 + 3.19e8T + 5.58e17T^{2} \)
11 \( 1 + 6.15e10iT - 7.40e21T^{2} \)
13 \( 1 - 1.45e11iT - 2.47e23T^{2} \)
17 \( 1 - 9.09e12T + 6.90e25T^{2} \)
19 \( 1 + 2.56e13iT - 7.14e26T^{2} \)
23 \( 1 + 3.28e14T + 3.94e28T^{2} \)
29 \( 1 + 6.87e14iT - 5.13e30T^{2} \)
31 \( 1 + 4.52e15T + 2.08e31T^{2} \)
37 \( 1 - 1.80e15iT - 8.55e32T^{2} \)
41 \( 1 - 1.56e17T + 7.38e33T^{2} \)
43 \( 1 + 1.27e17iT - 2.00e34T^{2} \)
47 \( 1 + 2.70e17T + 1.30e35T^{2} \)
53 \( 1 + 2.47e18iT - 1.62e36T^{2} \)
59 \( 1 + 2.19e18iT - 1.54e37T^{2} \)
61 \( 1 + 8.68e18iT - 3.10e37T^{2} \)
67 \( 1 + 1.31e19iT - 2.22e38T^{2} \)
71 \( 1 + 6.01e18T + 7.52e38T^{2} \)
73 \( 1 - 3.87e19T + 1.34e39T^{2} \)
79 \( 1 - 7.07e19T + 7.08e39T^{2} \)
83 \( 1 - 3.61e19iT - 1.99e40T^{2} \)
89 \( 1 + 4.11e20T + 8.65e40T^{2} \)
97 \( 1 - 1.14e20T + 5.27e41T^{2} \)
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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

−16.18067100937816858755597854390, −13.98943991797895955239308616438, −12.60902808004546149499539432523, −11.20288098261674712780738471017, −9.794379521186920413572544025022, −7.84775954178887330008022983490, −6.60478588955502255936838073409, −3.48506117085425750673643335886, −2.08138718502471786857944268555, −0.44885318386853354338823377590, 1.31383542567500311118074670413, 4.27565077636119377295026694621, 5.66857923970170734973390148090, 7.75110696797076194371454250840, 9.342271775696049105696273025714, 10.22764017323451720563145194071, 12.59428154556523315622898382835, 14.58515204456415471921063340343, 16.02212910676496161819784793634, 16.59531493055634677576395865007

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