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Properties

Label 2-20-4.3-c10-0-11
Degree $2$
Conductor $20$
Sign $0.868 - 0.495i$
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  + (30.9 − 8.19i)2-s + 206. i·3-s + (889. − 507. i)4-s + 1.39e3·5-s + (1.69e3 + 6.38e3i)6-s + 1.52e3i·7-s + (2.33e4 − 2.29e4i)8-s + 1.64e4·9-s + (4.32e4 − 1.14e4i)10-s + 2.94e5i·11-s + (1.04e5 + 1.83e5i)12-s + 1.98e5·13-s + (1.25e4 + 4.72e4i)14-s + 2.88e5i·15-s + (5.33e5 − 9.02e5i)16-s + 3.67e5·17-s + ⋯
L(s)  = 1  + (0.966 − 0.256i)2-s + 0.849i·3-s + (0.868 − 0.495i)4-s + 0.447·5-s + (0.217 + 0.821i)6-s + 0.0908i·7-s + (0.712 − 0.701i)8-s + 0.278·9-s + (0.432 − 0.114i)10-s + 1.82i·11-s + (0.420 + 0.737i)12-s + 0.533·13-s + (0.0232 + 0.0878i)14-s + 0.379i·15-s + (0.509 − 0.860i)16-s + 0.259·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.868 - 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.868 - 0.495i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ 0.868 - 0.495i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(3.51099 + 0.930731i\)
\(L(\frac12)\) \(\approx\) \(3.51099 + 0.930731i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-30.9 + 8.19i)T \)
5 \( 1 - 1.39e3T \)
good3 \( 1 - 206. iT - 5.90e4T^{2} \)
7 \( 1 - 1.52e3iT - 2.82e8T^{2} \)
11 \( 1 - 2.94e5iT - 2.59e10T^{2} \)
13 \( 1 - 1.98e5T + 1.37e11T^{2} \)
17 \( 1 - 3.67e5T + 2.01e12T^{2} \)
19 \( 1 + 9.19e5iT - 6.13e12T^{2} \)
23 \( 1 + 2.57e6iT - 4.14e13T^{2} \)
29 \( 1 + 1.40e7T + 4.20e14T^{2} \)
31 \( 1 + 4.90e7iT - 8.19e14T^{2} \)
37 \( 1 + 1.21e8T + 4.80e15T^{2} \)
41 \( 1 + 1.48e8T + 1.34e16T^{2} \)
43 \( 1 + 1.41e8iT - 2.16e16T^{2} \)
47 \( 1 + 3.99e7iT - 5.25e16T^{2} \)
53 \( 1 - 1.03e8T + 1.74e17T^{2} \)
59 \( 1 + 4.38e8iT - 5.11e17T^{2} \)
61 \( 1 + 2.83e8T + 7.13e17T^{2} \)
67 \( 1 - 2.00e9iT - 1.82e18T^{2} \)
71 \( 1 + 1.54e9iT - 3.25e18T^{2} \)
73 \( 1 - 3.44e9T + 4.29e18T^{2} \)
79 \( 1 + 2.88e9iT - 9.46e18T^{2} \)
83 \( 1 - 3.41e9iT - 1.55e19T^{2} \)
89 \( 1 + 2.47e9T + 3.11e19T^{2} \)
97 \( 1 + 2.70e9T + 7.37e19T^{2} \)
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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

−15.56283445948601943403360324452, −14.91112934595831952830856260341, −13.39217018846654533146848491665, −12.19348655384049382670165210042, −10.56287552785341571318259468487, −9.608349127055052081934816664605, −7.00801361164278204012483666536, −5.19527488036244092353097121483, −3.96520191669718888184357346281, −1.95408057269721732964558385819, 1.43855809114541097303097134092, 3.38201744623155003280579738995, 5.59681477590639849553631630172, 6.79489643354240927582414977066, 8.337747046829351555275274720987, 10.77426736145714056802475439644, 12.18314124356077552875394213056, 13.43790204275863599438365763119, 14.02606498326619507856520575536, 15.77229225099015358423718591902

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