Properties

Label 2-405-9.7-c3-0-28
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  + (−1.03 + 1.79i)2-s + (1.85 + 3.20i)4-s + (−2.5 − 4.33i)5-s + (2.33 − 4.03i)7-s − 24.2·8-s + 10.3·10-s + (−4.44 + 7.70i)11-s + (−17.3 − 29.9i)13-s + (4.83 + 8.36i)14-s + (10.3 − 17.9i)16-s + 2.66·17-s + 125.·19-s + (9.25 − 16.0i)20-s + (−9.21 − 15.9i)22-s + (−65.9 − 114. i)23-s + ⋯
L(s)  = 1  + (−0.366 + 0.634i)2-s + (0.231 + 0.400i)4-s + (−0.223 − 0.387i)5-s + (0.125 − 0.217i)7-s − 1.07·8-s + 0.327·10-s + (−0.121 + 0.211i)11-s + (−0.369 − 0.639i)13-s + (0.0922 + 0.159i)14-s + (0.161 − 0.279i)16-s + 0.0380·17-s + 1.51·19-s + (0.103 − 0.179i)20-s + (−0.0893 − 0.154i)22-s + (−0.598 − 1.03i)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: $\chi_{405} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.282985922\)
\(L(\frac12)\) \(\approx\) \(1.282985922\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.03 - 1.79i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-2.33 + 4.03i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (4.44 - 7.70i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (17.3 + 29.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 2.66T + 4.91e3T^{2} \)
19 \( 1 - 125.T + 6.85e3T^{2} \)
23 \( 1 + (65.9 + 114. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-35.6 + 61.6i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (6.71 + 11.6i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 283.T + 5.06e4T^{2} \)
41 \( 1 + (191. + 332. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-169. + 293. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (39.1 - 67.7i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 254.T + 1.48e5T^{2} \)
59 \( 1 + (-16.4 - 28.4i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (93.4 - 161. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (203. + 352. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 966.T + 3.57e5T^{2} \)
73 \( 1 - 276.T + 3.89e5T^{2} \)
79 \( 1 + (-573. + 993. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (89.0 - 154. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 806.T + 7.04e5T^{2} \)
97 \( 1 + (-619. + 1.07e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.75035021729526484457554241902, −9.714223600035032567548455725028, −8.797287799342636862666389681082, −7.84083169553229284120118280938, −7.36817024577935611218333458746, −6.18004546694946373878199711471, −5.12495904036175116898980973582, −3.79525189942613279740580099802, −2.53148285091781437053483612048, −0.55379698533793517296256809791, 1.12704334570821606898504555551, 2.43378700435334734170608993485, 3.51259685764783207292915434329, 5.08993953937233316520112055508, 6.09785381233313696305577344688, 7.14134006613753357172045694081, 8.176381758524050792674457766050, 9.454207672231262081064387853846, 9.835076019794852975101565557758, 10.99287423025038284778908381979

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