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Properties

Label 2-20-4.3-c4-0-4
Degree $2$
Conductor $20$
Sign $0.473 + 0.881i$
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  + (−2.05 + 3.43i)2-s − 15.5i·3-s + (−7.56 − 14.0i)4-s + 11.1·5-s + (53.4 + 31.9i)6-s − 37.6i·7-s + (63.9 + 2.96i)8-s − 161.·9-s + (−22.9 + 38.3i)10-s + 26.6i·11-s + (−219. + 117. i)12-s + 58.0·13-s + (129. + 77.2i)14-s − 174. i·15-s + (−141. + 213. i)16-s + 467.·17-s + ⋯
L(s)  = 1  + (−0.513 + 0.858i)2-s − 1.73i·3-s + (−0.473 − 0.881i)4-s + 0.447·5-s + (1.48 + 0.888i)6-s − 0.767i·7-s + (0.998 + 0.0462i)8-s − 1.99·9-s + (−0.229 + 0.383i)10-s + 0.220i·11-s + (−1.52 + 0.818i)12-s + 0.343·13-s + (0.658 + 0.394i)14-s − 0.774i·15-s + (−0.552 + 0.833i)16-s + 1.61·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 + 0.881i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.473 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.473 + 0.881i$
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: \((2,\ 20,\ (\ :2),\ 0.473 + 0.881i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.832946 - 0.498198i\)
\(L(\frac12)\) \(\approx\) \(0.832946 - 0.498198i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.05 - 3.43i)T \)
5 \( 1 - 11.1T \)
good3 \( 1 + 15.5iT - 81T^{2} \)
7 \( 1 + 37.6iT - 2.40e3T^{2} \)
11 \( 1 - 26.6iT - 1.46e4T^{2} \)
13 \( 1 - 58.0T + 2.85e4T^{2} \)
17 \( 1 - 467.T + 8.35e4T^{2} \)
19 \( 1 - 428. iT - 1.30e5T^{2} \)
23 \( 1 + 360. iT - 2.79e5T^{2} \)
29 \( 1 - 964.T + 7.07e5T^{2} \)
31 \( 1 + 417. iT - 9.23e5T^{2} \)
37 \( 1 + 1.79e3T + 1.87e6T^{2} \)
41 \( 1 + 469.T + 2.82e6T^{2} \)
43 \( 1 - 27.7iT - 3.41e6T^{2} \)
47 \( 1 - 1.53e3iT - 4.87e6T^{2} \)
53 \( 1 + 276.T + 7.89e6T^{2} \)
59 \( 1 - 3.81e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.05e3T + 1.38e7T^{2} \)
67 \( 1 - 1.16e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.68e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.00e3T + 2.83e7T^{2} \)
79 \( 1 + 705. iT - 3.89e7T^{2} \)
83 \( 1 + 1.62e3iT - 4.74e7T^{2} \)
89 \( 1 - 7.15e3T + 6.27e7T^{2} \)
97 \( 1 + 1.30e4T + 8.85e7T^{2} \)
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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

−17.45836553472469711312126721014, −16.55543158628798460437593188327, −14.40745335476681804497076761796, −13.69301969067342275827241252318, −12.33840581399422521849784393257, −10.25972452007220088929205565447, −8.268115878597598656524037078044, −7.18970592916696548478377162420, −5.92709740218944042590620142340, −1.21650811634170806680548546729, 3.23616120169063072411840900659, 5.16383608956449348785813032079, 8.652051372507505909245174491005, 9.658613660633761313207384048830, 10.68920802850046118809626356272, 11.99735332445735081295093600647, 13.94270822900395804580731601155, 15.48161930000001253522284897289, 16.56659720842344102860144713839, 17.71128471867471916579042144442

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