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Properties

Label 2-20-4.3-c4-0-3
Degree $2$
Conductor $20$
Sign $0.901 - 0.433i$
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  + (3.90 − 0.888i)2-s + 12.9i·3-s + (14.4 − 6.93i)4-s − 11.1·5-s + (11.5 + 50.6i)6-s − 78.0i·7-s + (50.0 − 39.8i)8-s − 87.7·9-s + (−43.6 + 9.93i)10-s + 95.6i·11-s + (90.0 + 187. i)12-s − 159.·13-s + (−69.3 − 304. i)14-s − 145. i·15-s + (159. − 199. i)16-s + 22.5·17-s + ⋯
L(s)  = 1  + (0.975 − 0.222i)2-s + 1.44i·3-s + (0.901 − 0.433i)4-s − 0.447·5-s + (0.320 + 1.40i)6-s − 1.59i·7-s + (0.782 − 0.622i)8-s − 1.08·9-s + (−0.436 + 0.0993i)10-s + 0.790i·11-s + (0.625 + 1.30i)12-s − 0.944·13-s + (−0.353 − 1.55i)14-s − 0.645i·15-s + (0.624 − 0.780i)16-s + 0.0778·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.901 - 0.433i$
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: \((2,\ 20,\ (\ :2),\ 0.901 - 0.433i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.89571 + 0.431893i\)
\(L(\frac12)\) \(\approx\) \(1.89571 + 0.431893i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.90 + 0.888i)T \)
5 \( 1 + 11.1T \)
good3 \( 1 - 12.9iT - 81T^{2} \)
7 \( 1 + 78.0iT - 2.40e3T^{2} \)
11 \( 1 - 95.6iT - 1.46e4T^{2} \)
13 \( 1 + 159.T + 2.85e4T^{2} \)
17 \( 1 - 22.5T + 8.35e4T^{2} \)
19 \( 1 - 324. iT - 1.30e5T^{2} \)
23 \( 1 + 204. iT - 2.79e5T^{2} \)
29 \( 1 - 295.T + 7.07e5T^{2} \)
31 \( 1 - 407. iT - 9.23e5T^{2} \)
37 \( 1 + 2.15e3T + 1.87e6T^{2} \)
41 \( 1 - 1.36e3T + 2.82e6T^{2} \)
43 \( 1 + 1.23e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.98e3iT - 4.87e6T^{2} \)
53 \( 1 - 4.59e3T + 7.89e6T^{2} \)
59 \( 1 - 1.38e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.65e3T + 1.38e7T^{2} \)
67 \( 1 + 8.93e3iT - 2.01e7T^{2} \)
71 \( 1 + 3.37e3iT - 2.54e7T^{2} \)
73 \( 1 + 1.46e3T + 2.83e7T^{2} \)
79 \( 1 - 6.30e3iT - 3.89e7T^{2} \)
83 \( 1 + 6.10e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.70e3T + 6.27e7T^{2} \)
97 \( 1 + 1.98e3T + 8.85e7T^{2} \)
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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

−17.05599536137608019969740348477, −16.14712465124558600066044022583, −14.98679495115036232446905096338, −14.05749154365881569790540607452, −12.31277309332503557924965307340, −10.69310563559640886524383494681, −10.00495194978395050077215569846, −7.26315889081305750293262555881, −4.79021516881876163455814255927, −3.78519168894219958666541151103, 2.53786627333733070441724890601, 5.55973673318911359404430185843, 7.01369392855813179967148391679, 8.434818032765931911695861642781, 11.59281023267816598200073538974, 12.28158717998790952109830749743, 13.35962594457699462093586566452, 14.73717915016450976162703535306, 15.89696150564899281168255374650, 17.56434928433278706976521739043

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