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Properties

Label 2-20-20.7-c5-0-5
Degree $2$
Conductor $20$
Sign $0.266 - 0.963i$
Analytic cond. $3.20767$
Root an. cond. $1.79099$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 + 4i)2-s + 32i·4-s + (38 + 41i)5-s + (−128 + 128i)8-s − 243i·9-s + (−12 + 316i)10-s + (719 − 719i)13-s − 1.02e3·16-s + (−717 − 717i)17-s + (972 − 972i)18-s + (−1.31e3 + 1.21e3i)20-s + (−237 + 3.11e3i)25-s + 5.75e3·26-s + 8.56e3i·29-s + (−4.09e3 − 4.09e3i)32-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + (0.679 + 0.733i)5-s + (−0.707 + 0.707i)8-s − i·9-s + (−0.0379 + 0.999i)10-s + (1.17 − 1.17i)13-s − 16-s + (−0.601 − 0.601i)17-s + (0.707 − 0.707i)18-s + (−0.733 + 0.679i)20-s + (−0.0758 + 0.997i)25-s + 1.66·26-s + 1.89i·29-s + (−0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.266 - 0.963i$
Analytic conductor: \(3.20767\)
Root analytic conductor: \(1.79099\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.63441 + 1.24379i\)
\(L(\frac12)\) \(\approx\) \(1.63441 + 1.24379i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 - 4i)T \)
5 \( 1 + (-38 - 41i)T \)
good3 \( 1 + 243iT^{2} \)
7 \( 1 - 1.68e4iT^{2} \)
11 \( 1 - 1.61e5T^{2} \)
13 \( 1 + (-719 + 719i)T - 3.71e5iT^{2} \)
17 \( 1 + (717 + 717i)T + 1.41e6iT^{2} \)
19 \( 1 + 2.47e6T^{2} \)
23 \( 1 + 6.43e6iT^{2} \)
29 \( 1 - 8.56e3iT - 2.05e7T^{2} \)
31 \( 1 - 2.86e7T^{2} \)
37 \( 1 + (1.17e4 + 1.17e4i)T + 6.93e7iT^{2} \)
41 \( 1 - 4.95e3T + 1.15e8T^{2} \)
43 \( 1 + 1.47e8iT^{2} \)
47 \( 1 - 2.29e8iT^{2} \)
53 \( 1 + (-2.37e4 + 2.37e4i)T - 4.18e8iT^{2} \)
59 \( 1 + 7.14e8T^{2} \)
61 \( 1 + 5.49e4T + 8.44e8T^{2} \)
67 \( 1 - 1.35e9iT^{2} \)
71 \( 1 - 1.80e9T^{2} \)
73 \( 1 + (3.43e4 - 3.43e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 3.07e9T^{2} \)
83 \( 1 + 3.93e9iT^{2} \)
89 \( 1 - 1.40e5iT - 5.58e9T^{2} \)
97 \( 1 + (-3.43e4 - 3.43e4i)T + 8.58e9iT^{2} \)
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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

−17.68042566346976468088043014887, −15.97153468610770875600758445752, −14.92479841635355311450675549347, −13.80684094386807822589105846901, −12.60433177603834056131724293569, −10.89269666887750239271734571355, −8.928426237167835127411821879286, −6.98708908057612980006958061687, −5.71670857638454933790124513986, −3.31923024199394702806445294093, 1.82464575287869434280692369639, 4.46409222812172831304990880757, 6.09864020149309213865293067008, 8.817869440260561836869181617063, 10.35347358724105618611397896637, 11.71503586664072084945421664084, 13.30080491769444111585650901095, 13.82992199286038374655354623587, 15.64122081029977962749113580978, 16.94057795811067793899836132142

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