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Properties

Label 2-2e3-8.3-c20-0-2
Degree $2$
Conductor $8$
Sign $0.625 - 0.779i$
Analytic cond. $20.2811$
Root an. cond. $4.50345$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−749. − 697. i)2-s − 1.61e4·3-s + (7.62e4 + 1.04e6i)4-s − 8.44e6i·5-s + (1.20e7 + 1.12e7i)6-s − 6.17e7i·7-s + (6.72e8 − 8.37e8i)8-s − 3.22e9·9-s + (−5.89e9 + 6.33e9i)10-s − 2.72e10·11-s + (−1.22e9 − 1.68e10i)12-s − 6.05e10i·13-s + (−4.30e10 + 4.62e10i)14-s + 1.36e11i·15-s + (−1.08e12 + 1.59e11i)16-s − 3.07e11·17-s + ⋯
L(s)  = 1  + (−0.732 − 0.680i)2-s − 0.272·3-s + (0.0726 + 0.997i)4-s − 0.865i·5-s + (0.199 + 0.185i)6-s − 0.218i·7-s + (0.625 − 0.779i)8-s − 0.925·9-s + (−0.589 + 0.633i)10-s − 1.05·11-s + (−0.0198 − 0.272i)12-s − 0.439i·13-s + (−0.148 + 0.160i)14-s + 0.236i·15-s + (−0.989 + 0.144i)16-s − 0.152·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.779i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.625 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.625 - 0.779i$
Analytic conductor: \(20.2811\)
Root analytic conductor: \(4.50345\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{8} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 8,\ (\ :10),\ 0.625 - 0.779i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.400597 + 0.192154i\)
\(L(\frac12)\) \(\approx\) \(0.400597 + 0.192154i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (749. + 697. i)T \)
good3 \( 1 + 1.61e4T + 3.48e9T^{2} \)
5 \( 1 + 8.44e6iT - 9.53e13T^{2} \)
7 \( 1 + 6.17e7iT - 7.97e16T^{2} \)
11 \( 1 + 2.72e10T + 6.72e20T^{2} \)
13 \( 1 + 6.05e10iT - 1.90e22T^{2} \)
17 \( 1 + 3.07e11T + 4.06e24T^{2} \)
19 \( 1 - 7.50e12T + 3.75e25T^{2} \)
23 \( 1 - 5.30e13iT - 1.71e27T^{2} \)
29 \( 1 - 5.67e14iT - 1.76e29T^{2} \)
31 \( 1 + 1.32e14iT - 6.71e29T^{2} \)
37 \( 1 + 2.44e14iT - 2.31e31T^{2} \)
41 \( 1 + 1.67e16T + 1.80e32T^{2} \)
43 \( 1 + 8.85e14T + 4.67e32T^{2} \)
47 \( 1 - 7.52e16iT - 2.76e33T^{2} \)
53 \( 1 - 2.98e17iT - 3.05e34T^{2} \)
59 \( 1 + 6.61e16T + 2.61e35T^{2} \)
61 \( 1 - 1.14e17iT - 5.08e35T^{2} \)
67 \( 1 - 1.03e18T + 3.32e36T^{2} \)
71 \( 1 + 4.52e18iT - 1.05e37T^{2} \)
73 \( 1 + 3.78e18T + 1.84e37T^{2} \)
79 \( 1 + 9.29e18iT - 8.96e37T^{2} \)
83 \( 1 - 2.33e19T + 2.40e38T^{2} \)
89 \( 1 + 4.76e19T + 9.72e38T^{2} \)
97 \( 1 + 7.53e19T + 5.43e39T^{2} \)
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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

−17.11437168320234740039171522455, −15.94729595176152213436961799377, −13.44671994362258546452838493593, −12.10887165600358609173234255549, −10.72105112753345130324376931802, −9.099415057950877882376875219501, −7.72807329601555137875676077816, −5.20285175056676615697437785803, −3.05520780793859666105405733149, −1.10756194422560305964128884921, 0.24415449313720473725729478692, 2.53498933115058389980863431139, 5.37195219441553213103570208298, 6.79870375391410752611156126578, 8.371305237770345024341380588679, 10.15635743479038945809274261610, 11.43329641172551746232700591394, 13.94036899814118855818757212150, 15.19163202468126302380410157047, 16.55882635514953864520783022449

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