↑

Properties

Label 2-810-45.29-c2-0-4
Degree $2$
Conductor $810$
Sign $0.425 - 0.905i$
Analytic cond. $22.0709$
Root an. cond. $4.69796$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (−2.48 − 4.33i)5-s + (−5.12 − 2.95i)7-s − 2.82·8-s + (−7.07 − 0.0209i)10-s + (−3.29 − 1.90i)11-s + (−19.2 + 11.1i)13-s + (−7.24 + 4.18i)14-s + (−2.00 + 3.46i)16-s − 1.20·17-s + 29.3·19-s + (−5.02 + 8.64i)20-s + (−4.66 + 2.69i)22-s + (8.07 + 13.9i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.497 − 0.867i)5-s + (−0.732 − 0.422i)7-s − 0.353·8-s + (−0.707 − 0.00209i)10-s + (−0.299 − 0.173i)11-s + (−1.48 + 0.854i)13-s + (−0.517 + 0.298i)14-s + (−0.125 + 0.216i)16-s − 0.0708·17-s + 1.54·19-s + (−0.251 + 0.432i)20-s + (−0.212 + 0.122i)22-s + (0.350 + 0.607i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.425 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.425 - 0.905i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ 0.425 - 0.905i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3733006347\)
\(L(\frac12)\) \(\approx\) \(0.3733006347\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
5 \( 1 + (2.48 + 4.33i)T \)
good7 \( 1 + (5.12 + 2.95i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (3.29 + 1.90i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (19.2 - 11.1i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 1.20T + 289T^{2} \)
19 \( 1 - 29.3T + 361T^{2} \)
23 \( 1 + (-8.07 - 13.9i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-39.2 - 22.6i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (21.8 + 37.9i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 48.4iT - 1.36e3T^{2} \)
41 \( 1 + (2.35 - 1.36i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (17.3 + 9.99i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-12.1 + 21.0i)T + (-1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 10.8T + 2.80e3T^{2} \)
59 \( 1 + (-16.7 + 9.69i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (9.19 - 15.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (71.9 - 41.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 121. iT - 5.04e3T^{2} \)
73 \( 1 + 105. iT - 5.32e3T^{2} \)
79 \( 1 + (47.1 - 81.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (47.3 - 81.9i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 132. iT - 7.92e3T^{2} \)
97 \( 1 + (103. + 59.5i)T + (4.70e3 + 8.14e3i)T^{2} \)
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

−9.956323911602925564887596898843, −9.651837623274107224472685451119, −8.694878638066548159170846486336, −7.56914777393781747134496366733, −6.83412024143063973825157025942, −5.42799181790958494686402247997, −4.77163611362733012776746781306, −3.76878137604270249499158339790, −2.75699395007725675459172580835, −1.21210599075496180795599322256, 0.11981625025612648658979359880, 2.70213648869578051529902988408, 3.21644311698340023833805838784, 4.61583293729446278005038668621, 5.51044067033722025548899796889, 6.49030909148715992952477212576, 7.34070651965750698297610120215, 7.81174148138317729780310741875, 9.025326034888738682777906355173, 9.930463815948192872381144880240

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