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Properties

Label 2-20-20.19-c4-0-1
Degree $2$
Conductor $20$
Sign $-0.842 - 0.538i$
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  + (1.35 + 3.76i)2-s − 13.9·3-s + (−12.3 + 10.2i)4-s + (7.64 + 23.8i)5-s + (−18.9 − 52.6i)6-s + 35.6·7-s + (−55.1 − 32.5i)8-s + 114.·9-s + (−79.2 + 61.0i)10-s + 169. i·11-s + (172. − 142. i)12-s − 72.8i·13-s + (48.3 + 134. i)14-s + (−107. − 333. i)15-s + (47.7 − 251. i)16-s + 25.2i·17-s + ⋯
L(s)  = 1  + (0.338 + 0.940i)2-s − 1.55·3-s + (−0.770 + 0.637i)4-s + (0.305 + 0.952i)5-s + (−0.527 − 1.46i)6-s + 0.728·7-s + (−0.861 − 0.508i)8-s + 1.41·9-s + (−0.792 + 0.610i)10-s + 1.40i·11-s + (1.19 − 0.991i)12-s − 0.430i·13-s + (0.246 + 0.685i)14-s + (−0.475 − 1.48i)15-s + (0.186 − 0.982i)16-s + 0.0872i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.842 - 0.538i$
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: \((2,\ 20,\ (\ :2),\ -0.842 - 0.538i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.239591 + 0.820315i\)
\(L(\frac12)\) \(\approx\) \(0.239591 + 0.820315i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 3.76i)T \)
5 \( 1 + (-7.64 - 23.8i)T \)
good3 \( 1 + 13.9T + 81T^{2} \)
7 \( 1 - 35.6T + 2.40e3T^{2} \)
11 \( 1 - 169. iT - 1.46e4T^{2} \)
13 \( 1 + 72.8iT - 2.85e4T^{2} \)
17 \( 1 - 25.2iT - 8.35e4T^{2} \)
19 \( 1 - 156. iT - 1.30e5T^{2} \)
23 \( 1 - 420.T + 2.79e5T^{2} \)
29 \( 1 - 439.T + 7.07e5T^{2} \)
31 \( 1 + 1.00e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.72e3iT - 1.87e6T^{2} \)
41 \( 1 - 2.18e3T + 2.82e6T^{2} \)
43 \( 1 - 180.T + 3.41e6T^{2} \)
47 \( 1 - 270.T + 4.87e6T^{2} \)
53 \( 1 + 924. iT - 7.89e6T^{2} \)
59 \( 1 + 4.12e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.49e3T + 1.38e7T^{2} \)
67 \( 1 - 860.T + 2.01e7T^{2} \)
71 \( 1 - 8.03e3iT - 2.54e7T^{2} \)
73 \( 1 + 5.77e3iT - 2.83e7T^{2} \)
79 \( 1 + 4.40e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.04e4T + 4.74e7T^{2} \)
89 \( 1 - 1.20e4T + 6.27e7T^{2} \)
97 \( 1 + 8.15e3iT - 8.85e7T^{2} \)
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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

−17.66902112890533443215334561370, −17.10311744228421752700911557596, −15.52522190110236570609722036844, −14.53325141334706823533087175520, −12.79952021522924396631740442063, −11.49684985062100234615696449291, −10.04843397159039642753367358831, −7.46180922645521559262092255964, −6.21049226341032687636370428370, −4.81101401637705946130155288369, 0.906416789560500028268067978590, 4.74206939418932135747582313116, 5.84346700778332811772151639386, 8.912128902450247789306040442187, 10.74710746475482523470929093413, 11.57305392511325906437196716505, 12.68133424568201804996341537356, 13.99901616463564967135555356523, 16.12344015044386826531293989467, 17.24608945226244733160936969751

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