↑

Properties

Label 2-160-20.3-c5-0-24
Degree $2$
Conductor $160$
Sign $-0.464 + 0.885i$
Analytic cond. $25.6614$
Root an. cond. $5.06570$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (2.56 − 2.56i)3-s + (3.91 + 55.7i)5-s + (−86.5 − 86.5i)7-s + 229. i·9-s − 138. i·11-s + (−393. − 393. i)13-s + (153. + 133. i)15-s + (724. − 724. i)17-s − 1.46e3·19-s − 444.·21-s + (2.85e3 − 2.85e3i)23-s + (−3.09e3 + 436. i)25-s + (1.21e3 + 1.21e3i)27-s − 2.51e3i·29-s − 2.96e3i·31-s + ⋯
L(s)  = 1  + (0.164 − 0.164i)3-s + (0.0700 + 0.997i)5-s + (−0.667 − 0.667i)7-s + 0.945i·9-s − 0.344i·11-s + (−0.645 − 0.645i)13-s + (0.175 + 0.152i)15-s + (0.607 − 0.607i)17-s − 0.930·19-s − 0.220·21-s + (1.12 − 1.12i)23-s + (−0.990 + 0.139i)25-s + (0.320 + 0.320i)27-s − 0.555i·29-s − 0.554i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.464 + 0.885i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ -0.464 + 0.885i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8156560583\)
\(L(\frac12)\) \(\approx\) \(0.8156560583\)
\(L(\frac{7}{2})\) 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 \( 1 + (-3.91 - 55.7i)T \)
good3 \( 1 + (-2.56 + 2.56i)T - 243iT^{2} \)
7 \( 1 + (86.5 + 86.5i)T + 1.68e4iT^{2} \)
11 \( 1 + 138. iT - 1.61e5T^{2} \)
13 \( 1 + (393. + 393. i)T + 3.71e5iT^{2} \)
17 \( 1 + (-724. + 724. i)T - 1.41e6iT^{2} \)
19 \( 1 + 1.46e3T + 2.47e6T^{2} \)
23 \( 1 + (-2.85e3 + 2.85e3i)T - 6.43e6iT^{2} \)
29 \( 1 + 2.51e3iT - 2.05e7T^{2} \)
31 \( 1 + 2.96e3iT - 2.86e7T^{2} \)
37 \( 1 + (1.45e3 - 1.45e3i)T - 6.93e7iT^{2} \)
41 \( 1 + 1.78e4T + 1.15e8T^{2} \)
43 \( 1 + (-4.66e3 + 4.66e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (2.07e4 + 2.07e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (1.15e4 + 1.15e4i)T + 4.18e8iT^{2} \)
59 \( 1 - 3.26e4T + 7.14e8T^{2} \)
61 \( 1 + 1.11e4T + 8.44e8T^{2} \)
67 \( 1 + (-1.53e4 - 1.53e4i)T + 1.35e9iT^{2} \)
71 \( 1 + 7.24e4iT - 1.80e9T^{2} \)
73 \( 1 + (6.69e3 + 6.69e3i)T + 2.07e9iT^{2} \)
79 \( 1 + 3.66e4T + 3.07e9T^{2} \)
83 \( 1 + (6.57e4 - 6.57e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 5.58e4iT - 5.58e9T^{2} \)
97 \( 1 + (-5.01e4 + 5.01e4i)T - 8.58e9iT^{2} \)
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

−11.50754791758988070754114652511, −10.47146169847996373942390145364, −9.985828397406564016606924500476, −8.376745888355268683311007438480, −7.32743953639608575107962086121, −6.52526080883700928541953357621, −5.05131364568093762243829185069, −3.42354600712298694334138133672, −2.36247244505374312128611754366, −0.25694696179726450777170363641, 1.50201025825136519431809960808, 3.22979400503280187967546450454, 4.58310199164533097544611447208, 5.80277561475471676658575323632, 6.95788030469057502282139432589, 8.486544177954759938957842529584, 9.264135138640689000532968110404, 9.935323088887391801706361068028, 11.55445611804316734004499806450, 12.51697109042690208936732268645

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