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Properties

Label 2-20-5.3-c6-0-1
Degree $2$
Conductor $20$
Sign $0.160 - 0.987i$
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  + (35.6 + 35.6i)3-s + (−115. + 47.8i)5-s + (53.9 − 53.9i)7-s + 1.81e3i·9-s + 1.89e3·11-s + (−954. − 954. i)13-s + (−5.81e3 − 2.41e3i)15-s + (3.80e3 − 3.80e3i)17-s − 2.61e3i·19-s + 3.84e3·21-s + (1.21e3 + 1.21e3i)23-s + (1.10e4 − 1.10e4i)25-s + (−3.85e4 + 3.85e4i)27-s − 2.68e4i·29-s + 4.63e3·31-s + ⋯
L(s)  = 1  + (1.31 + 1.31i)3-s + (−0.923 + 0.382i)5-s + (0.157 − 0.157i)7-s + 2.48i·9-s + 1.42·11-s + (−0.434 − 0.434i)13-s + (−1.72 − 0.714i)15-s + (0.774 − 0.774i)17-s − 0.381i·19-s + 0.415·21-s + (0.0994 + 0.0994i)23-s + (0.707 − 0.706i)25-s + (−1.95 + 1.95i)27-s − 1.09i·29-s + 0.155·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.160 - 0.987i$
Analytic conductor: \(4.60108\)
Root analytic conductor: \(2.14501\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3),\ 0.160 - 0.987i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.51638 + 1.28991i\)
\(L(\frac12)\) \(\approx\) \(1.51638 + 1.28991i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (115. - 47.8i)T \)
good3 \( 1 + (-35.6 - 35.6i)T + 729iT^{2} \)
7 \( 1 + (-53.9 + 53.9i)T - 1.17e5iT^{2} \)
11 \( 1 - 1.89e3T + 1.77e6T^{2} \)
13 \( 1 + (954. + 954. i)T + 4.82e6iT^{2} \)
17 \( 1 + (-3.80e3 + 3.80e3i)T - 2.41e7iT^{2} \)
19 \( 1 + 2.61e3iT - 4.70e7T^{2} \)
23 \( 1 + (-1.21e3 - 1.21e3i)T + 1.48e8iT^{2} \)
29 \( 1 + 2.68e4iT - 5.94e8T^{2} \)
31 \( 1 - 4.63e3T + 8.87e8T^{2} \)
37 \( 1 + (3.60e4 - 3.60e4i)T - 2.56e9iT^{2} \)
41 \( 1 - 4.05e4T + 4.75e9T^{2} \)
43 \( 1 + (7.46e4 + 7.46e4i)T + 6.32e9iT^{2} \)
47 \( 1 + (2.25e4 - 2.25e4i)T - 1.07e10iT^{2} \)
53 \( 1 + (-6.79e4 - 6.79e4i)T + 2.21e10iT^{2} \)
59 \( 1 - 3.53e4iT - 4.21e10T^{2} \)
61 \( 1 + 2.61e5T + 5.15e10T^{2} \)
67 \( 1 + (3.01e5 - 3.01e5i)T - 9.04e10iT^{2} \)
71 \( 1 - 2.74e5T + 1.28e11T^{2} \)
73 \( 1 + (4.83e4 + 4.83e4i)T + 1.51e11iT^{2} \)
79 \( 1 + 7.81e5iT - 2.43e11T^{2} \)
83 \( 1 + (4.32e5 + 4.32e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 4.80e5iT - 4.96e11T^{2} \)
97 \( 1 + (-1.65e5 + 1.65e5i)T - 8.32e11iT^{2} \)
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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

−16.89459708082855531777557186813, −15.66744329895768908235663992843, −14.81262325516551347348656063476, −13.93700919041574747490333391206, −11.69251582073325860782158764879, −10.19234193517100095724695637659, −8.946780404471435452579130159454, −7.58442482501633394335918211327, −4.43854831725143743427362802099, −3.16446211973865125079999109869, 1.43027752227882138817682003489, 3.62709448291690799297580286434, 6.83894170638430359115196384795, 8.094440440448956318841663911334, 9.128716687396614211397305208690, 11.88621432924239532954104194571, 12.65923929178798751087305946307, 14.22906304118343754722779443580, 14.92893566451124782720031417043, 16.80274954453709121501684598171

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