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Properties

Label 2-405-9.4-c3-0-14
Degree $2$
Conductor $405$
Sign $-0.984 + 0.173i$
Analytic cond. $23.8957$
Root an. cond. $4.88833$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.05 + 3.56i)2-s + (−4.47 + 7.75i)4-s + (−2.5 + 4.33i)5-s + (10.0 + 17.3i)7-s − 3.92·8-s − 20.5·10-s + (0.839 + 1.45i)11-s + (−5.55 + 9.62i)13-s + (−41.2 + 71.3i)14-s + (27.7 + 48.0i)16-s + 8.98·17-s − 50.6·19-s + (−22.3 − 38.7i)20-s + (−3.45 + 5.98i)22-s + (−107. + 185. i)23-s + ⋯
L(s)  = 1  + (0.727 + 1.26i)2-s + (−0.559 + 0.969i)4-s + (−0.223 + 0.387i)5-s + (0.540 + 0.936i)7-s − 0.173·8-s − 0.651·10-s + (0.0230 + 0.0398i)11-s + (−0.118 + 0.205i)13-s + (−0.786 + 1.36i)14-s + (0.433 + 0.750i)16-s + 0.128·17-s − 0.611·19-s + (−0.250 − 0.433i)20-s + (−0.0334 + 0.0579i)22-s + (−0.972 + 1.68i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.984 + 0.173i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -0.984 + 0.173i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.480399362\)
\(L(\frac12)\) \(\approx\) \(2.480399362\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 - 4.33i)T \)
good2 \( 1 + (-2.05 - 3.56i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-10.0 - 17.3i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-0.839 - 1.45i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (5.55 - 9.62i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 8.98T + 4.91e3T^{2} \)
19 \( 1 + 50.6T + 6.85e3T^{2} \)
23 \( 1 + (107. - 185. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-38.4 - 66.6i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-136. + 236. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 137.T + 5.06e4T^{2} \)
41 \( 1 + (-26.6 + 46.1i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (147. + 255. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-97.4 - 168. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 450.T + 1.48e5T^{2} \)
59 \( 1 + (240. - 416. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (337. + 585. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (447. - 774. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 721.T + 3.57e5T^{2} \)
73 \( 1 - 915.T + 3.89e5T^{2} \)
79 \( 1 + (29.8 + 51.6i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (371. + 643. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 1.54e3T + 7.04e5T^{2} \)
97 \( 1 + (-560. - 971. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.54905856452168630608850562113, −10.44007020199376309552239196310, −9.228419718941481854542337029860, −8.153710835750501312055628959400, −7.53772176834004868575968209132, −6.42677929855431086127321842633, −5.68572311383758234608195505344, −4.75914703143865397580787458610, −3.63856474968392028712751814739, −1.99627079106769046998292235380, 0.64512984861100292514085925163, 1.87841066702388538803981226599, 3.22396592514390922306725674005, 4.36423685815401636320074210649, 4.83935475928993756110710686698, 6.39234120278438526126853049790, 7.69587022989421724584499732389, 8.527828163160136679751517885419, 9.955078063108502790465557337829, 10.56655880848060784871640784698

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