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Properties

Label 2-20-20.3-c9-0-5
Degree $2$
Conductor $20$
Sign $0.498 - 0.866i$
Analytic cond. $10.3007$
Root an. cond. $3.20947$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.74 − 22.3i)2-s + (−68.1 + 68.1i)3-s + (−483. − 167. i)4-s + (−497. − 1.30e3i)5-s + (1.26e3 + 1.77e3i)6-s + (2.74e3 + 2.74e3i)7-s + (−5.53e3 + 1.01e4i)8-s + 1.04e4i·9-s + (−3.10e4 + 6.21e3i)10-s + 4.20e4i·11-s + (4.43e4 − 2.15e4i)12-s + (4.82e4 + 4.82e4i)13-s + (7.15e4 − 5.09e4i)14-s + (1.22e5 + 5.50e4i)15-s + (2.06e5 + 1.61e5i)16-s + (2.69e4 − 2.69e4i)17-s + ⋯
L(s)  = 1  + (0.165 − 0.986i)2-s + (−0.485 + 0.485i)3-s + (−0.945 − 0.326i)4-s + (−0.355 − 0.934i)5-s + (0.398 + 0.559i)6-s + (0.432 + 0.432i)7-s + (−0.478 + 0.878i)8-s + 0.528i·9-s + (−0.980 + 0.196i)10-s + 0.865i·11-s + (0.617 − 0.300i)12-s + (0.469 + 0.469i)13-s + (0.497 − 0.354i)14-s + (0.626 + 0.280i)15-s + (0.787 + 0.616i)16-s + (0.0783 − 0.0783i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.498 - 0.866i$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :9/2),\ 0.498 - 0.866i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.666237 + 0.385403i\)
\(L(\frac12)\) \(\approx\) \(0.666237 + 0.385403i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.74 + 22.3i)T \)
5 \( 1 + (497. + 1.30e3i)T \)
good3 \( 1 + (68.1 - 68.1i)T - 1.96e4iT^{2} \)
7 \( 1 + (-2.74e3 - 2.74e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 4.20e4iT - 2.35e9T^{2} \)
13 \( 1 + (-4.82e4 - 4.82e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-2.69e4 + 2.69e4i)T - 1.18e11iT^{2} \)
19 \( 1 + 3.57e5T + 3.22e11T^{2} \)
23 \( 1 + (7.98e5 - 7.98e5i)T - 1.80e12iT^{2} \)
29 \( 1 - 5.48e6iT - 1.45e13T^{2} \)
31 \( 1 + 8.10e6iT - 2.64e13T^{2} \)
37 \( 1 + (1.32e7 - 1.32e7i)T - 1.29e14iT^{2} \)
41 \( 1 - 7.49e6T + 3.27e14T^{2} \)
43 \( 1 + (2.51e7 - 2.51e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (-1.82e7 - 1.82e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (-3.95e6 - 3.95e6i)T + 3.29e15iT^{2} \)
59 \( 1 - 1.22e8T + 8.66e15T^{2} \)
61 \( 1 - 3.05e7T + 1.16e16T^{2} \)
67 \( 1 + (1.72e8 + 1.72e8i)T + 2.72e16iT^{2} \)
71 \( 1 - 2.97e8iT - 4.58e16T^{2} \)
73 \( 1 + (2.23e7 + 2.23e7i)T + 5.88e16iT^{2} \)
79 \( 1 - 3.86e8T + 1.19e17T^{2} \)
83 \( 1 + (1.87e8 - 1.87e8i)T - 1.86e17iT^{2} \)
89 \( 1 - 2.80e8iT - 3.50e17T^{2} \)
97 \( 1 + (6.97e8 - 6.97e8i)T - 7.60e17iT^{2} \)
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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

−16.51373645844125816654334028528, −15.09485564208337794700939824352, −13.43000557604129423288221272940, −12.16619153602231547765581804366, −11.18420171286063310038628437643, −9.726527543520128895348710656480, −8.328345093208816672680486876854, −5.30439010278525555830385348289, −4.25090904269944482252869093761, −1.73273295711171585058085237102, 0.37225677062567501378147995489, 3.72893901632023514171857249074, 5.91983625838223957366933088447, 7.02995968869729023714910836398, 8.414884781864637926074411393196, 10.57733237429034037141534023612, 12.09122283765700362327496099787, 13.64262397005232376750053535925, 14.72385264395557088945223155461, 15.90904619514211487110245587094

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