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Properties

Label 2-20-20.19-c10-0-17
Degree $2$
Conductor $20$
Sign $-0.999 + 0.0135i$
Analytic cond. $12.7071$
Root an. cond. $3.56470$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−28.1 − 15.1i)2-s − 349.·3-s + (565. + 853. i)4-s + (1.69e3 − 2.62e3i)5-s + (9.85e3 + 5.29e3i)6-s + 1.54e4·7-s + (−3.01e3 − 3.26e4i)8-s + 6.31e4·9-s + (−8.74e4 + 4.85e4i)10-s − 1.36e5i·11-s + (−1.97e5 − 2.98e5i)12-s + 4.67e5i·13-s + (−4.34e5 − 2.33e5i)14-s + (−5.90e5 + 9.18e5i)15-s + (−4.09e5 + 9.65e5i)16-s − 1.19e6i·17-s + ⋯
L(s)  = 1  + (−0.880 − 0.473i)2-s − 1.43·3-s + (0.552 + 0.833i)4-s + (0.540 − 0.841i)5-s + (1.26 + 0.680i)6-s + 0.916·7-s + (−0.0918 − 0.995i)8-s + 1.06·9-s + (−0.874 + 0.485i)10-s − 0.849i·11-s + (−0.794 − 1.19i)12-s + 1.25i·13-s + (−0.807 − 0.433i)14-s + (−0.778 + 1.21i)15-s + (−0.390 + 0.920i)16-s − 0.839i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0135i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.999 + 0.0135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.999 + 0.0135i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ -0.999 + 0.0135i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.00253275 - 0.374700i\)
\(L(\frac12)\) \(\approx\) \(0.00253275 - 0.374700i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (28.1 + 15.1i)T \)
5 \( 1 + (-1.69e3 + 2.62e3i)T \)
good3 \( 1 + 349.T + 5.90e4T^{2} \)
7 \( 1 - 1.54e4T + 2.82e8T^{2} \)
11 \( 1 + 1.36e5iT - 2.59e10T^{2} \)
13 \( 1 - 4.67e5iT - 1.37e11T^{2} \)
17 \( 1 + 1.19e6iT - 2.01e12T^{2} \)
19 \( 1 + 1.42e6iT - 6.13e12T^{2} \)
23 \( 1 + 1.04e7T + 4.14e13T^{2} \)
29 \( 1 - 8.98e6T + 4.20e14T^{2} \)
31 \( 1 + 2.79e7iT - 8.19e14T^{2} \)
37 \( 1 - 1.06e8iT - 4.80e15T^{2} \)
41 \( 1 + 1.66e8T + 1.34e16T^{2} \)
43 \( 1 + 7.53e7T + 2.16e16T^{2} \)
47 \( 1 + 2.17e8T + 5.25e16T^{2} \)
53 \( 1 + 5.32e8iT - 1.74e17T^{2} \)
59 \( 1 - 2.43e6iT - 5.11e17T^{2} \)
61 \( 1 + 1.38e9T + 7.13e17T^{2} \)
67 \( 1 - 1.26e9T + 1.82e18T^{2} \)
71 \( 1 - 2.18e9iT - 3.25e18T^{2} \)
73 \( 1 + 5.11e8iT - 4.29e18T^{2} \)
79 \( 1 + 4.53e9iT - 9.46e18T^{2} \)
83 \( 1 - 8.51e8T + 1.55e19T^{2} \)
89 \( 1 - 2.46e9T + 3.11e19T^{2} \)
97 \( 1 + 8.02e9iT - 7.37e19T^{2} \)
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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

−16.13364438966405795222409522478, −13.61639600736103932379277291877, −11.87579861541883878195942488044, −11.42860671508866445496589538801, −9.882900244116601153501105902881, −8.389739289397122526277509672387, −6.40063415874094204830214529038, −4.75715092247040066576711716079, −1.62818316082178859796839614310, −0.26249655550096966818769665317, 1.66595871547418022270288555200, 5.31188040517445825811373957554, 6.38300063706879507806156150486, 7.87141518237630790127308854828, 10.16694401761585870771043898102, 10.76800534094175369693360510701, 12.13959590145450789922867948806, 14.39664407814995848688971107466, 15.52315783159023183786031495043, 17.00481147163456370484960081314

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