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Properties

Label 2-20-20.19-c4-0-8
Degree $2$
Conductor $20$
Sign $0.693 + 0.720i$
Analytic cond. $2.06739$
Root an. cond. $1.43784$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.85 − 2.79i)2-s + 6.64·3-s + (0.324 − 15.9i)4-s + (−17.6 + 17.7i)5-s + (18.9 − 18.5i)6-s + 39.0·7-s + (−43.8 − 46.6i)8-s − 36.8·9-s + (−0.852 + 99.9i)10-s + 138. i·11-s + (2.15 − 106. i)12-s + 124. i·13-s + (111. − 109. i)14-s + (−117. + 117. i)15-s + (−255. − 10.3i)16-s − 160. i·17-s + ⋯
L(s)  = 1  + (0.714 − 0.699i)2-s + 0.737·3-s + (0.0202 − 0.999i)4-s + (−0.705 + 0.708i)5-s + (0.527 − 0.516i)6-s + 0.797·7-s + (−0.685 − 0.728i)8-s − 0.455·9-s + (−0.00852 + 0.999i)10-s + 1.14i·11-s + (0.0149 − 0.737i)12-s + 0.739i·13-s + (0.569 − 0.558i)14-s + (−0.520 + 0.522i)15-s + (−0.999 − 0.0405i)16-s − 0.555i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.693 + 0.720i$
Analytic conductor: \(2.06739\)
Root analytic conductor: \(1.43784\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :2),\ 0.693 + 0.720i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.75909 - 0.747956i\)
\(L(\frac12)\) \(\approx\) \(1.75909 - 0.747956i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.85 + 2.79i)T \)
5 \( 1 + (17.6 - 17.7i)T \)
good3 \( 1 - 6.64T + 81T^{2} \)
7 \( 1 - 39.0T + 2.40e3T^{2} \)
11 \( 1 - 138. iT - 1.46e4T^{2} \)
13 \( 1 - 124. iT - 2.85e4T^{2} \)
17 \( 1 + 160. iT - 8.35e4T^{2} \)
19 \( 1 + 650. iT - 1.30e5T^{2} \)
23 \( 1 - 416.T + 2.79e5T^{2} \)
29 \( 1 - 236.T + 7.07e5T^{2} \)
31 \( 1 + 41.5iT - 9.23e5T^{2} \)
37 \( 1 - 206. iT - 1.87e6T^{2} \)
41 \( 1 + 1.81e3T + 2.82e6T^{2} \)
43 \( 1 - 3.16e3T + 3.41e6T^{2} \)
47 \( 1 - 823.T + 4.87e6T^{2} \)
53 \( 1 - 4.86e3iT - 7.89e6T^{2} \)
59 \( 1 - 3.63e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.13e3T + 1.38e7T^{2} \)
67 \( 1 + 3.20e3T + 2.01e7T^{2} \)
71 \( 1 + 456. iT - 2.54e7T^{2} \)
73 \( 1 + 5.90e3iT - 2.83e7T^{2} \)
79 \( 1 + 2.80e3iT - 3.89e7T^{2} \)
83 \( 1 - 3.04e3T + 4.74e7T^{2} \)
89 \( 1 + 5.14e3T + 6.27e7T^{2} \)
97 \( 1 - 4.50e3iT - 8.85e7T^{2} \)
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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

−17.75430081595818990279713779832, −15.41500551566344977428088402897, −14.68818233250028804413771941144, −13.67579388265061921220331432045, −11.91728910520261210052009898053, −10.95617966011513640006096386636, −9.134393500822173186743280287245, −7.15206290104993692667242290314, −4.56274245629865773477901483105, −2.63073956649002068379589412787, 3.59490169026865388685655648431, 5.51984101383819057562775649313, 7.992657353597182222904771291196, 8.526770907975268528885542841105, 11.36433162645836682991820767289, 12.72869711081596799515104545806, 14.08456154910620734848519455758, 14.97934548282696329943260222085, 16.25844791121655207940127164962, 17.32491252460893008357047379724

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