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Properties

Label 2-405-9.4-c3-0-46
Degree $2$
Conductor $405$
Sign $-0.173 - 0.984i$
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  + (−1.06 − 1.84i)2-s + (1.73 − 3.01i)4-s + (−2.5 + 4.33i)5-s + (−15.3 − 26.5i)7-s − 24.4·8-s + 10.6·10-s + (−25.0 − 43.4i)11-s + (7.97 − 13.8i)13-s + (−32.6 + 56.5i)14-s + (12.0 + 20.8i)16-s + 105.·17-s − 21.3·19-s + (8.69 + 15.0i)20-s + (−53.3 + 92.3i)22-s + (−68.0 + 117. i)23-s + ⋯
L(s)  = 1  + (−0.375 − 0.650i)2-s + (0.217 − 0.376i)4-s + (−0.223 + 0.387i)5-s + (−0.828 − 1.43i)7-s − 1.07·8-s + 0.336·10-s + (−0.687 − 1.19i)11-s + (0.170 − 0.294i)13-s + (−0.623 + 1.07i)14-s + (0.187 + 0.325i)16-s + 1.50·17-s − 0.257·19-s + (0.0972 + 0.168i)20-s + (−0.516 + 0.894i)22-s + (−0.617 + 1.06i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.173 - 0.984i$
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: \((2,\ 405,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3397232284\)
\(L(\frac12)\) \(\approx\) \(0.3397232284\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.06 + 1.84i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (15.3 + 26.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (25.0 + 43.4i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-7.97 + 13.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 105.T + 4.91e3T^{2} \)
19 \( 1 + 21.3T + 6.85e3T^{2} \)
23 \( 1 + (68.0 - 117. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-112. - 194. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-112. + 195. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 416.T + 5.06e4T^{2} \)
41 \( 1 + (38.0 - 65.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (15.8 + 27.4i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (30.4 + 52.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 466.T + 1.48e5T^{2} \)
59 \( 1 + (47.7 - 82.6i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-178. - 309. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (43.9 - 76.0i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 412.T + 3.57e5T^{2} \)
73 \( 1 + 331.T + 3.89e5T^{2} \)
79 \( 1 + (-124. - 214. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-276. - 478. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 291.T + 7.04e5T^{2} \)
97 \( 1 + (99.3 + 171. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.30690706833671472318852721042, −9.714291329891039984793178269572, −8.362936192003674943416039538634, −7.39372677246556294748267984128, −6.37998070158727592958835409544, −5.45561204288180215502115786392, −3.60283005806627806457985529169, −3.05705475293480062842552897615, −1.18969607481503369864633214036, −0.13642427571285517820669134909, 2.25073774080615322608996484684, 3.31529663861330577177185364737, 4.92480281325066858510860014159, 5.99866924765844935303258100149, 6.80815545161374175141891435310, 7.949942607822487085205696065728, 8.553142785021766654431662707062, 9.497748435478666975905759415554, 10.27380121663108030255104884418, 11.98860966019694314547683224953

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