↑

Properties

Label 2-405-45.7-c2-0-32
Degree $2$
Conductor $405$
Sign $-0.735 + 0.677i$
Analytic cond. $11.0354$
Root an. cond. $3.32196$
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  + (−2.53 + 0.678i)2-s + (2.48 − 1.43i)4-s + (−1.78 − 4.67i)5-s + (−2.47 + 0.662i)7-s + (2.09 − 2.09i)8-s + (7.68 + 10.6i)10-s + (1.02 − 1.78i)11-s + (22.3 + 5.98i)13-s + (5.81 − 3.35i)14-s + (−9.62 + 16.6i)16-s + (−14.0 − 14.0i)17-s − 7.13i·19-s + (−11.1 − 9.05i)20-s + (−1.39 + 5.21i)22-s + (−17.1 − 4.60i)23-s + ⋯
L(s)  = 1  + (−1.26 + 0.339i)2-s + (0.621 − 0.358i)4-s + (−0.356 − 0.934i)5-s + (−0.353 + 0.0946i)7-s + (0.261 − 0.261i)8-s + (0.768 + 1.06i)10-s + (0.0935 − 0.162i)11-s + (1.71 + 0.460i)13-s + (0.415 − 0.239i)14-s + (−0.601 + 1.04i)16-s + (−0.826 − 0.826i)17-s − 0.375i·19-s + (−0.557 − 0.452i)20-s + (−0.0634 + 0.236i)22-s + (−0.746 − 0.200i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.735 + 0.677i$
Analytic conductor: \(11.0354\)
Root analytic conductor: \(3.32196\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1),\ -0.735 + 0.677i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.116099 - 0.297688i\)
\(L(\frac12)\) \(\approx\) \(0.116099 - 0.297688i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.78 + 4.67i)T \)
good2 \( 1 + (2.53 - 0.678i)T + (3.46 - 2i)T^{2} \)
7 \( 1 + (2.47 - 0.662i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (-1.02 + 1.78i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-22.3 - 5.98i)T + (146. + 84.5i)T^{2} \)
17 \( 1 + (14.0 + 14.0i)T + 289iT^{2} \)
19 \( 1 + 7.13iT - 361T^{2} \)
23 \( 1 + (17.1 + 4.60i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + (-31.6 - 18.2i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-11.0 - 19.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (39.3 + 39.3i)T + 1.36e3iT^{2} \)
41 \( 1 + (7.34 + 12.7i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (17.1 + 64.1i)T + (-1.60e3 + 924.5i)T^{2} \)
47 \( 1 + (76.8 - 20.6i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-21.4 + 21.4i)T - 2.80e3iT^{2} \)
59 \( 1 + (31.9 - 18.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.1 + 40.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-28.1 + 105. i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 102.T + 5.04e3T^{2} \)
73 \( 1 + (10.4 - 10.4i)T - 5.32e3iT^{2} \)
79 \( 1 + (63.6 + 36.7i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (20.7 + 77.4i)T + (-5.96e3 + 3.44e3i)T^{2} \)
89 \( 1 - 73.0iT - 7.92e3T^{2} \)
97 \( 1 + (107. - 28.6i)T + (8.14e3 - 4.70e3i)T^{2} \)
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

−10.53047728174923301503600374145, −9.430025529780976505834214941298, −8.721292512962893161971463960393, −8.365152791770497340403726767032, −7.09110307133625677947830995339, −6.26975903922401927274996674092, −4.79691759093903908722727083409, −3.65464146050845988621157974819, −1.54375744206273334254477848419, −0.22828043673095453336425034987, 1.52313496934781563090393016902, 3.02363705145546480566249759703, 4.21962750335212228922218487679, 6.06392116156250795032310369890, 6.82142603255875091293853037148, 8.222317068617300230910375284475, 8.368167607677532841294674831559, 9.854428867821026120110956008101, 10.28801581469248218343250458802, 11.17557093820871674841093253873

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