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Properties

Label 2-20-20.7-c5-0-2
Degree $2$
Conductor $20$
Sign $-0.690 - 0.723i$
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  + (−5.05 + 2.53i)2-s + (10.7 + 10.7i)3-s + (19.1 − 25.6i)4-s + (−36.9 + 41.9i)5-s + (−81.4 − 26.9i)6-s + (−129. + 129. i)7-s + (−31.4 + 178. i)8-s − 13.1i·9-s + (80.4 − 305. i)10-s + 299. i·11-s + (480. − 70.2i)12-s + (370. − 370. i)13-s + (326. − 984. i)14-s + (−845. + 53.1i)15-s + (−293. − 981. i)16-s + (1.42e3 + 1.42e3i)17-s + ⋯
L(s)  = 1  + (−0.893 + 0.448i)2-s + (0.687 + 0.687i)3-s + (0.597 − 0.802i)4-s + (−0.661 + 0.750i)5-s + (−0.923 − 0.306i)6-s + (−0.999 + 0.999i)7-s + (−0.173 + 0.984i)8-s − 0.0541i·9-s + (0.254 − 0.967i)10-s + 0.746i·11-s + (0.962 − 0.140i)12-s + (0.608 − 0.608i)13-s + (0.444 − 1.34i)14-s + (−0.970 + 0.0610i)15-s + (−0.286 − 0.958i)16-s + (1.19 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.690 - 0.723i$
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: \((2,\ 20,\ (\ :5/2),\ -0.690 - 0.723i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.346205 + 0.808488i\)
\(L(\frac12)\) \(\approx\) \(0.346205 + 0.808488i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.05 - 2.53i)T \)
5 \( 1 + (36.9 - 41.9i)T \)
good3 \( 1 + (-10.7 - 10.7i)T + 243iT^{2} \)
7 \( 1 + (129. - 129. i)T - 1.68e4iT^{2} \)
11 \( 1 - 299. iT - 1.61e5T^{2} \)
13 \( 1 + (-370. + 370. i)T - 3.71e5iT^{2} \)
17 \( 1 + (-1.42e3 - 1.42e3i)T + 1.41e6iT^{2} \)
19 \( 1 - 540.T + 2.47e6T^{2} \)
23 \( 1 + (-167. - 167. i)T + 6.43e6iT^{2} \)
29 \( 1 + 973. iT - 2.05e7T^{2} \)
31 \( 1 - 2.92e3iT - 2.86e7T^{2} \)
37 \( 1 + (-1.64e3 - 1.64e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.01e4T + 1.15e8T^{2} \)
43 \( 1 + (4.73e3 + 4.73e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (1.60e4 - 1.60e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (1.44e4 - 1.44e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 2.29e4T + 7.14e8T^{2} \)
61 \( 1 + 8.05e3T + 8.44e8T^{2} \)
67 \( 1 + (2.71e4 - 2.71e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 5.71e4iT - 1.80e9T^{2} \)
73 \( 1 + (-5.29e4 + 5.29e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 8.73e4T + 3.07e9T^{2} \)
83 \( 1 + (6.66e4 + 6.66e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 3.17e4iT - 5.58e9T^{2} \)
97 \( 1 + (-6.23e4 - 6.23e4i)T + 8.58e9iT^{2} \)
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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

−17.92075544844923885608722730248, −16.11195950972480150583702697839, −15.32568906174865204463188210596, −14.64648002239991767432994942277, −12.23900206116678507347715438890, −10.42819655129788193452072464197, −9.384805938137917090284954330509, −8.011514761474525075206003058818, −6.22147543859337752845320025058, −3.16689346263562032547632396025, 0.849473786078770918174902028452, 3.41462834980612193570901275465, 7.12709975727782488704281449611, 8.233470400318345274804464549320, 9.600345986370893414042358110154, 11.35300261891084265333226148149, 12.79861549551857909563913033597, 13.78357339945082658868496425578, 16.24700088240568623201601036453, 16.54941749361117080403138718254

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