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Properties

Label 2-20-5.3-c10-0-1
Degree $2$
Conductor $20$
Sign $-0.761 - 0.648i$
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  + (186. + 186. i)3-s + (−473. + 3.08e3i)5-s + (−7.25e3 + 7.25e3i)7-s + 1.04e4i·9-s − 3.22e4·11-s + (−1.23e5 − 1.23e5i)13-s + (−6.63e5 + 4.87e5i)15-s + (−1.34e6 + 1.34e6i)17-s + 3.80e6i·19-s − 2.70e6·21-s + (−1.48e6 − 1.48e6i)23-s + (−9.31e6 − 2.92e6i)25-s + (9.06e6 − 9.06e6i)27-s + 1.97e7i·29-s + 3.84e7·31-s + ⋯
L(s)  = 1  + (0.766 + 0.766i)3-s + (−0.151 + 0.988i)5-s + (−0.431 + 0.431i)7-s + 0.176i·9-s − 0.200·11-s + (−0.332 − 0.332i)13-s + (−0.874 + 0.641i)15-s + (−0.947 + 0.947i)17-s + 1.53i·19-s − 0.661·21-s + (−0.230 − 0.230i)23-s + (−0.954 − 0.299i)25-s + (0.631 − 0.631i)27-s + 0.963i·29-s + 1.34·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.761 - 0.648i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ -0.761 - 0.648i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.568141 + 1.54266i\)
\(L(\frac12)\) \(\approx\) \(0.568141 + 1.54266i\)
\(L(6)\) 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 \( 1 + (473. - 3.08e3i)T \)
good3 \( 1 + (-186. - 186. i)T + 5.90e4iT^{2} \)
7 \( 1 + (7.25e3 - 7.25e3i)T - 2.82e8iT^{2} \)
11 \( 1 + 3.22e4T + 2.59e10T^{2} \)
13 \( 1 + (1.23e5 + 1.23e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (1.34e6 - 1.34e6i)T - 2.01e12iT^{2} \)
19 \( 1 - 3.80e6iT - 6.13e12T^{2} \)
23 \( 1 + (1.48e6 + 1.48e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 1.97e7iT - 4.20e14T^{2} \)
31 \( 1 - 3.84e7T + 8.19e14T^{2} \)
37 \( 1 + (-6.70e7 + 6.70e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 1.28e8T + 1.34e16T^{2} \)
43 \( 1 + (3.08e7 + 3.08e7i)T + 2.16e16iT^{2} \)
47 \( 1 + (1.99e8 - 1.99e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (-3.65e8 - 3.65e8i)T + 1.74e17iT^{2} \)
59 \( 1 + 1.07e9iT - 5.11e17T^{2} \)
61 \( 1 - 2.62e8T + 7.13e17T^{2} \)
67 \( 1 + (5.30e8 - 5.30e8i)T - 1.82e18iT^{2} \)
71 \( 1 - 2.34e9T + 3.25e18T^{2} \)
73 \( 1 + (-1.92e9 - 1.92e9i)T + 4.29e18iT^{2} \)
79 \( 1 + 3.63e9iT - 9.46e18T^{2} \)
83 \( 1 + (1.23e9 + 1.23e9i)T + 1.55e19iT^{2} \)
89 \( 1 - 9.01e9iT - 3.11e19T^{2} \)
97 \( 1 + (9.05e9 - 9.05e9i)T - 7.37e19iT^{2} \)
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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

−16.01586383735841846832565774060, −15.06831356766698352667502352953, −14.25203975904716243440921395883, −12.49324184935994743625217417394, −10.71256222620951578625906871920, −9.633085400940796456360660367645, −8.126329989304146389859349125206, −6.25358883697008138147833508289, −3.92077118920240471857004232328, −2.62078721352958959184758762899, 0.63038750875085807939759737741, 2.46832017581561429175349480866, 4.65185746941958984072164437948, 6.94339302973738197276587987610, 8.273176509770167663158161169929, 9.529525612942780174511638082498, 11.62478024943763917257818629510, 13.18480260281193080531088187241, 13.62674032840443117764120182900, 15.46510987509701661614812206496

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