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Properties

Label 2-2e3-8.5-c21-0-13
Degree $2$
Conductor $8$
Sign $0.946 + 0.322i$
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  + (582. + 1.32e3i)2-s − 2.64e4i·3-s + (−1.41e6 + 1.54e6i)4-s − 7.17e6i·5-s + (3.50e7 − 1.54e7i)6-s − 5.46e8·7-s + (−2.87e9 − 9.78e8i)8-s + 9.76e9·9-s + (9.51e9 − 4.18e9i)10-s + 4.39e9i·11-s + (4.08e10 + 3.74e10i)12-s − 7.95e11i·13-s + (−3.18e11 − 7.24e11i)14-s − 1.89e11·15-s + (−3.78e11 − 4.38e12i)16-s + 5.71e12·17-s + ⋯
L(s)  = 1  + (0.402 + 0.915i)2-s − 0.258i·3-s + (−0.675 + 0.736i)4-s − 0.328i·5-s + (0.236 − 0.104i)6-s − 0.731·7-s + (−0.946 − 0.322i)8-s + 0.933·9-s + (0.300 − 0.132i)10-s + 0.0511i·11-s + (0.190 + 0.174i)12-s − 1.60i·13-s + (−0.294 − 0.669i)14-s − 0.0849·15-s + (−0.0861 − 0.996i)16-s + 0.687·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8\)    =    \(2^{3}\)
Sign: $0.946 + 0.322i$
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: \((2,\ 8,\ (\ :21/2),\ 0.946 + 0.322i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.77793 - 0.294229i\)
\(L(\frac12)\) \(\approx\) \(1.77793 - 0.294229i\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-582. - 1.32e3i)T \)
good3 \( 1 + 2.64e4iT - 1.04e10T^{2} \)
5 \( 1 + 7.17e6iT - 4.76e14T^{2} \)
7 \( 1 + 5.46e8T + 5.58e17T^{2} \)
11 \( 1 - 4.39e9iT - 7.40e21T^{2} \)
13 \( 1 + 7.95e11iT - 2.47e23T^{2} \)
17 \( 1 - 5.71e12T + 6.90e25T^{2} \)
19 \( 1 + 4.93e12iT - 7.14e26T^{2} \)
23 \( 1 - 1.23e14T + 3.94e28T^{2} \)
29 \( 1 + 1.25e15iT - 5.13e30T^{2} \)
31 \( 1 - 4.10e15T + 2.08e31T^{2} \)
37 \( 1 + 8.37e15iT - 8.55e32T^{2} \)
41 \( 1 + 1.06e17T + 7.38e33T^{2} \)
43 \( 1 + 9.16e16iT - 2.00e34T^{2} \)
47 \( 1 + 4.35e17T + 1.30e35T^{2} \)
53 \( 1 + 1.27e18iT - 1.62e36T^{2} \)
59 \( 1 + 6.91e18iT - 1.54e37T^{2} \)
61 \( 1 + 4.23e18iT - 3.10e37T^{2} \)
67 \( 1 - 1.61e19iT - 2.22e38T^{2} \)
71 \( 1 + 2.72e19T + 7.52e38T^{2} \)
73 \( 1 + 1.06e19T + 1.34e39T^{2} \)
79 \( 1 - 1.17e20T + 7.08e39T^{2} \)
83 \( 1 - 2.07e20iT - 1.99e40T^{2} \)
89 \( 1 - 4.42e20T + 8.65e40T^{2} \)
97 \( 1 + 1.41e20T + 5.27e41T^{2} \)
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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

−16.18613124807772988517104578078, −15.10916402300739301396212850257, −13.27627021999087027824957404899, −12.51039098659871268389329499629, −9.883451420808286705754123114308, −8.096614101553302117082360843094, −6.68380871052449277797570561548, −5.08929804002160578409922011210, −3.29182904670518513590118910755, −0.60982382395866700855842140223, 1.40416858649289743574633129917, 3.18425762524818531671121977624, 4.59457364107744703441017398871, 6.62636629829022822798080249230, 9.266024435766431822852656238442, 10.41264500851337383634610244999, 11.99001656118777191990189616308, 13.36580890889115907614170966868, 14.78307403085019264540236726941, 16.41448617924465254278510130552

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