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Properties

Label 2-20-4.3-c4-0-0
Degree $2$
Conductor $20$
Sign $-0.794 - 0.607i$
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  + (−1.28 + 3.78i)2-s + 8.14i·3-s + (−12.7 − 9.71i)4-s − 11.1·5-s + (−30.8 − 10.4i)6-s + 63.6i·7-s + (53.1 − 35.7i)8-s + 14.7·9-s + (14.3 − 42.3i)10-s + 33.3i·11-s + (79.0 − 103. i)12-s + 274.·13-s + (−240. − 81.5i)14-s − 91.0i·15-s + (67.2 + 247. i)16-s − 284.·17-s + ⋯
L(s)  = 1  + (−0.320 + 0.947i)2-s + 0.904i·3-s + (−0.794 − 0.607i)4-s − 0.447·5-s + (−0.856 − 0.289i)6-s + 1.29i·7-s + (0.829 − 0.558i)8-s + 0.181·9-s + (0.143 − 0.423i)10-s + 0.275i·11-s + (0.549 − 0.718i)12-s + 1.62·13-s + (−1.22 − 0.416i)14-s − 0.404i·15-s + (0.262 + 0.964i)16-s − 0.985·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.794 - 0.607i$
Analytic conductor: \(2.06739\)
Root analytic conductor: \(1.43784\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :2),\ -0.794 - 0.607i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.303225 + 0.896170i\)
\(L(\frac12)\) \(\approx\) \(0.303225 + 0.896170i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 - 3.78i)T \)
5 \( 1 + 11.1T \)
good3 \( 1 - 8.14iT - 81T^{2} \)
7 \( 1 - 63.6iT - 2.40e3T^{2} \)
11 \( 1 - 33.3iT - 1.46e4T^{2} \)
13 \( 1 - 274.T + 2.85e4T^{2} \)
17 \( 1 + 284.T + 8.35e4T^{2} \)
19 \( 1 - 5.17iT - 1.30e5T^{2} \)
23 \( 1 + 584. iT - 2.79e5T^{2} \)
29 \( 1 - 344.T + 7.07e5T^{2} \)
31 \( 1 + 1.46e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.93e3T + 1.87e6T^{2} \)
41 \( 1 + 976.T + 2.82e6T^{2} \)
43 \( 1 - 2.04e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.56e3iT - 4.87e6T^{2} \)
53 \( 1 - 121.T + 7.89e6T^{2} \)
59 \( 1 + 3.45e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.13e3T + 1.38e7T^{2} \)
67 \( 1 + 1.69e3iT - 2.01e7T^{2} \)
71 \( 1 + 7.64e3iT - 2.54e7T^{2} \)
73 \( 1 + 6.00e3T + 2.83e7T^{2} \)
79 \( 1 - 4.01e3iT - 3.89e7T^{2} \)
83 \( 1 + 3.01e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.19e3T + 6.27e7T^{2} \)
97 \( 1 + 3.02e3T + 8.85e7T^{2} \)
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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

−18.08160950395874654978953136322, −16.33754077045236681739920359830, −15.61690300486954745142083517162, −14.87168136714347233510654574938, −13.03472661203570775991722335589, −11.04429553952683334569598522115, −9.432840398400645968971823971512, −8.355507038398017634366126364043, −6.20718987913644133587998449760, −4.42382911741089768897641314942, 1.10994083720626196515603962918, 3.90418165575209924187002198514, 7.09037036744047384980235459054, 8.489229471122626971265170884479, 10.46956325339476094179146429371, 11.57797153922715693410980801633, 13.15778640555512484953722021291, 13.70680782356246451985865512459, 16.06674239338214142081662826378, 17.54007173321370918880931195460

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