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Properties

Label 2-20-20.19-c6-0-12
Degree $2$
Conductor $20$
Sign $0.355 + 0.934i$
Analytic cond. $4.60108$
Root an. cond. $2.14501$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.68 − 7.09i)2-s + 31.7·3-s + (−36.7 − 52.3i)4-s + (121. + 30.8i)5-s + (117. − 225. i)6-s − 88.8·7-s + (−507. + 68.1i)8-s + 279.·9-s + (665. − 746. i)10-s − 1.79e3i·11-s + (−1.16e3 − 1.66e3i)12-s + 3.32e3i·13-s + (−327. + 630. i)14-s + (3.84e3 + 980. i)15-s + (−1.38e3 + 3.85e3i)16-s + 3.00e3i·17-s + ⋯
L(s)  = 1  + (0.460 − 0.887i)2-s + 1.17·3-s + (−0.574 − 0.818i)4-s + (0.969 + 0.246i)5-s + (0.542 − 1.04i)6-s − 0.258·7-s + (−0.991 + 0.133i)8-s + 0.384·9-s + (0.665 − 0.746i)10-s − 1.34i·11-s + (−0.676 − 0.962i)12-s + 1.51i·13-s + (−0.119 + 0.229i)14-s + (1.14 + 0.290i)15-s + (−0.338 + 0.940i)16-s + 0.610i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.355 + 0.934i$
Analytic conductor: \(4.60108\)
Root analytic conductor: \(2.14501\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3),\ 0.355 + 0.934i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.13097 - 1.46987i\)
\(L(\frac12)\) \(\approx\) \(2.13097 - 1.46987i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.68 + 7.09i)T \)
5 \( 1 + (-121. - 30.8i)T \)
good3 \( 1 - 31.7T + 729T^{2} \)
7 \( 1 + 88.8T + 1.17e5T^{2} \)
11 \( 1 + 1.79e3iT - 1.77e6T^{2} \)
13 \( 1 - 3.32e3iT - 4.82e6T^{2} \)
17 \( 1 - 3.00e3iT - 2.41e7T^{2} \)
19 \( 1 - 6.87e3iT - 4.70e7T^{2} \)
23 \( 1 - 5.90e3T + 1.48e8T^{2} \)
29 \( 1 + 1.40e4T + 5.94e8T^{2} \)
31 \( 1 + 5.12e4iT - 8.87e8T^{2} \)
37 \( 1 + 1.89e4iT - 2.56e9T^{2} \)
41 \( 1 + 1.17e4T + 4.75e9T^{2} \)
43 \( 1 + 1.00e5T + 6.32e9T^{2} \)
47 \( 1 - 1.29e5T + 1.07e10T^{2} \)
53 \( 1 + 4.75e4iT - 2.21e10T^{2} \)
59 \( 1 + 1.39e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.99e5T + 5.15e10T^{2} \)
67 \( 1 + 4.08e4T + 9.04e10T^{2} \)
71 \( 1 - 2.22e5iT - 1.28e11T^{2} \)
73 \( 1 + 6.33e5iT - 1.51e11T^{2} \)
79 \( 1 + 1.66e5iT - 2.43e11T^{2} \)
83 \( 1 + 3.28e5T + 3.26e11T^{2} \)
89 \( 1 - 2.77e4T + 4.96e11T^{2} \)
97 \( 1 + 2.20e5iT - 8.32e11T^{2} \)
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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

−16.79497127902170575834899648265, −14.75887822175431022877904353595, −13.97224491526178996218584096479, −13.18278505571084138344042965296, −11.30445861062782603190193255256, −9.743568772847808715383584180288, −8.724268300317685571470668361193, −6.02480992615854354042232035308, −3.56611174177796565880454159296, −2.01414654888662185632572771505, 2.86867471187780736265125447813, 5.15017256177608461003309326191, 7.11706665422227275967652922152, 8.651660466378835761212390497747, 9.794459378707065902380450517810, 12.69345832289773668894847979690, 13.51786441302560249631425642882, 14.68106884770257387416672662063, 15.56954181214510871285384505494, 17.25217720886132106243706716598

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