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Properties

Label 2-20-20.3-c9-0-17
Degree $2$
Conductor $20$
Sign $0.999 - 0.0253i$
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  + (21.1 + 7.98i)2-s + (175. − 175. i)3-s + (384. + 338. i)4-s + (−373. + 1.34e3i)5-s + (5.10e3 − 2.30e3i)6-s + (4.88e3 + 4.88e3i)7-s + (5.43e3 + 1.02e4i)8-s − 4.16e4i·9-s + (−1.86e4 + 2.55e4i)10-s − 6.39e4i·11-s + (1.26e5 − 8.09e3i)12-s + (−5.24e4 − 5.24e4i)13-s + (6.44e4 + 1.42e5i)14-s + (1.70e5 + 3.01e5i)15-s + (3.34e4 + 2.60e5i)16-s + (8.35e3 − 8.35e3i)17-s + ⋯
L(s)  = 1  + (0.935 + 0.352i)2-s + (1.24 − 1.24i)3-s + (0.750 + 0.660i)4-s + (−0.267 + 0.963i)5-s + (1.60 − 0.727i)6-s + (0.769 + 0.769i)7-s + (0.469 + 0.882i)8-s − 2.11i·9-s + (−0.590 + 0.807i)10-s − 1.31i·11-s + (1.76 − 0.112i)12-s + (−0.509 − 0.509i)13-s + (0.448 + 0.991i)14-s + (0.869 + 1.53i)15-s + (0.127 + 0.991i)16-s + (0.0242 − 0.0242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0253i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.999 - 0.0253i$
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: \((2,\ 20,\ (\ :9/2),\ 0.999 - 0.0253i)\)

Particular Values

\(L(5)\) \(\approx\) \(4.27315 + 0.0542486i\)
\(L(\frac12)\) \(\approx\) \(4.27315 + 0.0542486i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-21.1 - 7.98i)T \)
5 \( 1 + (373. - 1.34e3i)T \)
good3 \( 1 + (-175. + 175. i)T - 1.96e4iT^{2} \)
7 \( 1 + (-4.88e3 - 4.88e3i)T + 4.03e7iT^{2} \)
11 \( 1 + 6.39e4iT - 2.35e9T^{2} \)
13 \( 1 + (5.24e4 + 5.24e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-8.35e3 + 8.35e3i)T - 1.18e11iT^{2} \)
19 \( 1 + 4.19e5T + 3.22e11T^{2} \)
23 \( 1 + (1.10e6 - 1.10e6i)T - 1.80e12iT^{2} \)
29 \( 1 - 2.13e6iT - 1.45e13T^{2} \)
31 \( 1 + 1.18e6iT - 2.64e13T^{2} \)
37 \( 1 + (2.41e6 - 2.41e6i)T - 1.29e14iT^{2} \)
41 \( 1 - 1.43e7T + 3.27e14T^{2} \)
43 \( 1 + (-7.84e6 + 7.84e6i)T - 5.02e14iT^{2} \)
47 \( 1 + (-2.79e7 - 2.79e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (7.80e7 + 7.80e7i)T + 3.29e15iT^{2} \)
59 \( 1 + 7.94e7T + 8.66e15T^{2} \)
61 \( 1 + 2.25e6T + 1.16e16T^{2} \)
67 \( 1 + (-4.54e7 - 4.54e7i)T + 2.72e16iT^{2} \)
71 \( 1 - 1.58e7iT - 4.58e16T^{2} \)
73 \( 1 + (-8.51e7 - 8.51e7i)T + 5.88e16iT^{2} \)
79 \( 1 - 3.29e8T + 1.19e17T^{2} \)
83 \( 1 + (2.05e8 - 2.05e8i)T - 1.86e17iT^{2} \)
89 \( 1 + 6.70e8iT - 3.50e17T^{2} \)
97 \( 1 + (-9.32e8 + 9.32e8i)T - 7.60e17iT^{2} \)
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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

−15.51566713366303262988639578005, −14.53189164374086167523525226375, −13.88912705174547801497998454619, −12.54749471117166057496812578805, −11.31873146619816640199694162317, −8.424303054807153331897812977923, −7.54232522169028104477469158929, −6.07334170510483955876674148419, −3.31272517765249392037559674506, −2.17790478198564858317510639681, 2.05330093023170467072388954020, 4.19981974350524410053409939800, 4.63712253499757386487528623040, 7.76330011018381057902655971331, 9.438910634942045760581559809472, 10.61641547625667784206535166049, 12.38107489047049385689860949075, 13.87762323552870966606241651070, 14.74402023650275196955796322769, 15.71470437651339937147991626627

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