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Properties

Label 2-20-20.3-c1-0-0
Degree $2$
Conductor $20$
Sign $0.850 - 0.525i$
Analytic cond. $0.159700$
Root an. cond. $0.399625$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s − 2i·4-s + (−2 − i)5-s + (2 + 2i)8-s + 3i·9-s + (3 − i)10-s + (−1 − i)13-s − 4·16-s + (3 − 3i)17-s + (−3 − 3i)18-s + (−2 + 4i)20-s + (3 + 4i)25-s + 2·26-s − 4i·29-s + (4 − 4i)32-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − i·4-s + (−0.894 − 0.447i)5-s + (0.707 + 0.707i)8-s + i·9-s + (0.948 − 0.316i)10-s + (−0.277 − 0.277i)13-s − 16-s + (0.727 − 0.727i)17-s + (−0.707 − 0.707i)18-s + (−0.447 + 0.894i)20-s + (0.600 + 0.800i)25-s + 0.392·26-s − 0.742i·29-s + (0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(0.159700\)
Root analytic conductor: \(0.399625\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.412523 + 0.117189i\)
\(L(\frac12)\) \(\approx\) \(0.412523 + 0.117189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 + (2 + i)T \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 + (-3 + 3i)T - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (7 - 7i)T - 37iT^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-9 - 9i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (11 + 11i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 + (-13 + 13i)T - 97iT^{2} \)
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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

−18.63950642032327279407698589487, −17.05197663834121013206113169629, −16.18889154393944071424592053479, −15.12087754482813165213151877280, −13.61776821454158890200258592321, −11.72375060446644028152469300011, −10.19606565450750423970831368888, −8.476557285997462789360043827500, −7.36427401884056470781626500805, −5.09388682361178306352765485359, 3.63351761276323569547339575571, 7.08937157793967758816517109516, 8.640379848934142593144264292186, 10.22889472215849124414124733521, 11.62341391771111472288252712239, 12.57162116369725768064984496401, 14.60863390629778476709057892335, 15.98851315466574025753781465568, 17.36278766896153774645413833010, 18.54530994742234557988936814298

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