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Properties

Label 2-20-5.2-c10-0-3
Degree $2$
Conductor $20$
Sign $0.594 + 0.803i$
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  + (−189. + 189. i)3-s + (−3.11e3 + 261. i)5-s + (−3.93e3 − 3.93e3i)7-s − 1.28e4i·9-s + 1.15e5·11-s + (3.02e5 − 3.02e5i)13-s + (5.40e5 − 6.39e5i)15-s + (−5.38e5 − 5.38e5i)17-s − 1.70e6i·19-s + 1.49e6·21-s + (6.91e6 − 6.91e6i)23-s + (9.62e6 − 1.62e6i)25-s + (−8.76e6 − 8.76e6i)27-s + 3.34e7i·29-s − 4.11e7·31-s + ⋯
L(s)  = 1  + (−0.780 + 0.780i)3-s + (−0.996 + 0.0835i)5-s + (−0.234 − 0.234i)7-s − 0.216i·9-s + 0.716·11-s + (0.814 − 0.814i)13-s + (0.712 − 0.842i)15-s + (−0.379 − 0.379i)17-s − 0.688i·19-s + 0.365·21-s + (1.07 − 1.07i)23-s + (0.986 − 0.166i)25-s + (−0.610 − 0.610i)27-s + 1.63i·29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.594 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.594 + 0.803i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ 0.594 + 0.803i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.682214 - 0.343771i\)
\(L(\frac12)\) \(\approx\) \(0.682214 - 0.343771i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (3.11e3 - 261. i)T \)
good3 \( 1 + (189. - 189. i)T - 5.90e4iT^{2} \)
7 \( 1 + (3.93e3 + 3.93e3i)T + 2.82e8iT^{2} \)
11 \( 1 - 1.15e5T + 2.59e10T^{2} \)
13 \( 1 + (-3.02e5 + 3.02e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (5.38e5 + 5.38e5i)T + 2.01e12iT^{2} \)
19 \( 1 + 1.70e6iT - 6.13e12T^{2} \)
23 \( 1 + (-6.91e6 + 6.91e6i)T - 4.14e13iT^{2} \)
29 \( 1 - 3.34e7iT - 4.20e14T^{2} \)
31 \( 1 + 4.11e7T + 8.19e14T^{2} \)
37 \( 1 + (1.73e7 + 1.73e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 1.92e7T + 1.34e16T^{2} \)
43 \( 1 + (-2.41e7 + 2.41e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (1.09e8 + 1.09e8i)T + 5.25e16iT^{2} \)
53 \( 1 + (-3.10e8 + 3.10e8i)T - 1.74e17iT^{2} \)
59 \( 1 + 9.72e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.48e9T + 7.13e17T^{2} \)
67 \( 1 + (1.43e9 + 1.43e9i)T + 1.82e18iT^{2} \)
71 \( 1 + 1.28e9T + 3.25e18T^{2} \)
73 \( 1 + (7.60e8 - 7.60e8i)T - 4.29e18iT^{2} \)
79 \( 1 - 1.16e9iT - 9.46e18T^{2} \)
83 \( 1 + (-3.50e9 + 3.50e9i)T - 1.55e19iT^{2} \)
89 \( 1 - 6.77e9iT - 3.11e19T^{2} \)
97 \( 1 + (1.13e10 + 1.13e10i)T + 7.37e19iT^{2} \)
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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

−15.94972420128232286434434343263, −14.78129710925599746555370305919, −12.89633529039463616041750829186, −11.36509622549725524958904033356, −10.61968629085409701637893269841, −8.787833603809065115633844745450, −6.92261397926982195373309186870, −5.03033558967459987576769638388, −3.59219576748000359630212151025, −0.42789620375465772299793732348, 1.23145173231071497504236615553, 3.89037482291335301218123343623, 6.04537292984008433694511501765, 7.30416493147197484236988584657, 9.001871888302661295683525063904, 11.23372266015059117951747483048, 11.96661274037527873424211548367, 13.18624708912759307407185876223, 14.94525990194839647506502861314, 16.26287423813236351907918212611

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