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Properties

Label 2-405-9.4-c3-0-0
Degree $2$
Conductor $405$
Sign $-0.766 + 0.642i$
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  + (1.5 + 2.59i)2-s + (−0.5 + 0.866i)4-s + (−2.5 + 4.33i)5-s + (−10 − 17.3i)7-s + 21·8-s − 15.0·10-s + (−12 − 20.7i)11-s + (−37 + 64.0i)13-s + (30.0 − 51.9i)14-s + (35.5 + 61.4i)16-s − 54·17-s − 124·19-s + (−2.50 − 4.33i)20-s + (36 − 62.3i)22-s + (−60 + 103. i)23-s + ⋯
L(s)  = 1  + (0.530 + 0.918i)2-s + (−0.0625 + 0.108i)4-s + (−0.223 + 0.387i)5-s + (−0.539 − 0.935i)7-s + 0.928·8-s − 0.474·10-s + (−0.328 − 0.569i)11-s + (−0.789 + 1.36i)13-s + (0.572 − 0.991i)14-s + (0.554 + 0.960i)16-s − 0.770·17-s − 1.49·19-s + (−0.0279 − 0.0484i)20-s + (0.348 − 0.604i)22-s + (−0.543 + 0.942i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.766 + 0.642i$
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: \((2,\ 405,\ (\ :3/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2606265749\)
\(L(\frac12)\) \(\approx\) \(0.2606265749\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.5 - 2.59i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (10 + 17.3i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (12 + 20.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (37 - 64.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 54T + 4.91e3T^{2} \)
19 \( 1 + 124T + 6.85e3T^{2} \)
23 \( 1 + (60 - 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (39 + 67.5i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (100 - 173. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 70T + 5.06e4T^{2} \)
41 \( 1 + (-165 + 285. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (46 + 79.6i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (12 + 20.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 450T + 1.48e5T^{2} \)
59 \( 1 + (-12 + 20.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-161 - 278. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-98 + 169. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 288T + 3.57e5T^{2} \)
73 \( 1 + 430T + 3.89e5T^{2} \)
79 \( 1 + (-260 - 450. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-78 - 135. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + (-143 - 247. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.18303315101428155919600767706, −10.60625082666576979044393584679, −9.611074289259951913285329527889, −8.385124213191721458265821864365, −7.18501394090863956526213688252, −6.83736563576361869336833637348, −5.85252641730685520321638595983, −4.55799364927326886866482286452, −3.78891379514766127224634154792, −2.00714366595282885093777767642, 0.06496620381125067126056767674, 2.11242072752000680220912541076, 2.85461931378549547883133816363, 4.20199065820092469585221511174, 5.08218184704548379492150143983, 6.29591604985339958467559572528, 7.61130234789460945148318755989, 8.449895574288421968419349383735, 9.603326435861890208073927206481, 10.46958143442891965539922343725

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