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Properties

Label 2-405-9.4-c3-0-4
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

\(L(2)\) \(\approx\) \(0.4179052145\)
\(L(\frac12)\) \(\approx\) \(0.4179052145\)
\(L(\frac{5}{2})\) 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

−10.64242540460174877212074903316, −10.17102159868620546083330937479, −9.321144161951093357792947208657, −8.602992565046768360185899042139, −7.05442837896064095777398280237, −6.44220325310688351622540790764, −4.86925689177323443874517668201, −3.78692971906753789824311146710, −2.34432299216424244253065808139, −1.23865926332527884056524629186, 0.17161051601563216432188918097, 2.50272903311968799131390611449, 3.45900330764592845205483565608, 5.48357609648767074543920883998, 6.00094508133569028804001667864, 7.03042738772054743014911304056, 7.939304051508737282863222268446, 8.769200870634848716516728820025, 9.589098157379380178436480150357, 10.50052817743630538045656477602

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