↑

Properties

Label 2-20-20.19-c8-0-16
Degree $2$
Conductor $20$
Sign $0.939 - 0.341i$
Analytic cond. $8.14757$
Root an. cond. $2.85439$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (14.9 + 5.76i)2-s + 76.0·3-s + (189. + 172. i)4-s + (291. − 552. i)5-s + (1.13e3 + 438. i)6-s + 269.·7-s + (1.83e3 + 3.66e3i)8-s − 779.·9-s + (7.53e3 − 6.57e3i)10-s − 4.92e3i·11-s + (1.44e4 + 1.30e4i)12-s + 2.44e4i·13-s + (4.02e3 + 1.55e3i)14-s + (2.21e4 − 4.20e4i)15-s + (6.34e3 + 6.52e4i)16-s + 6.22e4i·17-s + ⋯
L(s)  = 1  + (0.932 + 0.360i)2-s + 0.938·3-s + (0.740 + 0.672i)4-s + (0.466 − 0.884i)5-s + (0.875 + 0.338i)6-s + 0.112·7-s + (0.448 + 0.893i)8-s − 0.118·9-s + (0.753 − 0.657i)10-s − 0.336i·11-s + (0.695 + 0.630i)12-s + 0.855i·13-s + (0.104 + 0.0404i)14-s + (0.437 − 0.830i)15-s + (0.0967 + 0.995i)16-s + 0.744i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.939 - 0.341i$
Analytic conductor: \(8.14757\)
Root analytic conductor: \(2.85439\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :4),\ 0.939 - 0.341i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(3.75463 + 0.661406i\)
\(L(\frac12)\) \(\approx\) \(3.75463 + 0.661406i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-14.9 - 5.76i)T \)
5 \( 1 + (-291. + 552. i)T \)
good3 \( 1 - 76.0T + 6.56e3T^{2} \)
7 \( 1 - 269.T + 5.76e6T^{2} \)
11 \( 1 + 4.92e3iT - 2.14e8T^{2} \)
13 \( 1 - 2.44e4iT - 8.15e8T^{2} \)
17 \( 1 - 6.22e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.40e5iT - 1.69e10T^{2} \)
23 \( 1 + 2.90e5T + 7.83e10T^{2} \)
29 \( 1 + 8.15e5T + 5.00e11T^{2} \)
31 \( 1 + 1.57e6iT - 8.52e11T^{2} \)
37 \( 1 - 2.27e6iT - 3.51e12T^{2} \)
41 \( 1 - 4.85e6T + 7.98e12T^{2} \)
43 \( 1 - 3.94e6T + 1.16e13T^{2} \)
47 \( 1 - 2.95e5T + 2.38e13T^{2} \)
53 \( 1 + 4.51e6iT - 6.22e13T^{2} \)
59 \( 1 - 2.25e7iT - 1.46e14T^{2} \)
61 \( 1 - 9.61e6T + 1.91e14T^{2} \)
67 \( 1 - 8.33e6T + 4.06e14T^{2} \)
71 \( 1 + 2.78e7iT - 6.45e14T^{2} \)
73 \( 1 - 2.56e7iT - 8.06e14T^{2} \)
79 \( 1 + 2.31e7iT - 1.51e15T^{2} \)
83 \( 1 - 3.83e7T + 2.25e15T^{2} \)
89 \( 1 + 4.36e6T + 3.93e15T^{2} \)
97 \( 1 - 1.02e8iT - 7.83e15T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.32527355359862966298750999863, −14.95456300328791473558411570170, −13.86751448914021574188891885788, −13.03821626082611369820198928011, −11.49345427276432971103152263796, −9.188217557843356669079335137323, −7.951131433545126782105551740174, −5.92797942046613210015557925215, −4.15772413715755806073690165321, −2.21578279517380867433419913408, 2.20007424089621703475803876324, 3.48663783958303969523772373012, 5.75782154599097373783783567641, 7.54985904140351193666807489303, 9.695810116344962033273955206623, 10.99713308503726797287190982052, 12.64997524123768136925317860211, 14.12229239602758869913358135163, 14.50439472660312680897830556752, 15.84413614064598480279158172081

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