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Properties

Label 2-405-5.4-c3-0-48
Degree $2$
Conductor $405$
Sign $-0.695 + 0.718i$
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  − 1.85i·2-s + 4.55·4-s + (−7.77 + 8.03i)5-s − 1.02i·7-s − 23.3i·8-s + (14.9 + 14.4i)10-s − 16.1·11-s + 3.05i·13-s − 1.90·14-s − 6.86·16-s − 69.2i·17-s + 12.6·19-s + (−35.4 + 36.5i)20-s + 29.9i·22-s − 56.6i·23-s + ⋯
L(s)  = 1  − 0.656i·2-s + 0.568·4-s + (−0.695 + 0.718i)5-s − 0.0552i·7-s − 1.03i·8-s + (0.471 + 0.456i)10-s − 0.442·11-s + 0.0651i·13-s − 0.0363·14-s − 0.107·16-s − 0.988i·17-s + 0.152·19-s + (−0.395 + 0.408i)20-s + 0.290i·22-s − 0.513i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.695 + 0.718i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (244, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -0.695 + 0.718i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.360702422\)
\(L(\frac12)\) \(\approx\) \(1.360702422\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.77 - 8.03i)T \)
good2 \( 1 + 1.85iT - 8T^{2} \)
7 \( 1 + 1.02iT - 343T^{2} \)
11 \( 1 + 16.1T + 1.33e3T^{2} \)
13 \( 1 - 3.05iT - 2.19e3T^{2} \)
17 \( 1 + 69.2iT - 4.91e3T^{2} \)
19 \( 1 - 12.6T + 6.85e3T^{2} \)
23 \( 1 + 56.6iT - 1.21e4T^{2} \)
29 \( 1 + 196.T + 2.43e4T^{2} \)
31 \( 1 - 213.T + 2.97e4T^{2} \)
37 \( 1 + 299. iT - 5.06e4T^{2} \)
41 \( 1 + 139.T + 6.89e4T^{2} \)
43 \( 1 + 475. iT - 7.95e4T^{2} \)
47 \( 1 + 193. iT - 1.03e5T^{2} \)
53 \( 1 + 214. iT - 1.48e5T^{2} \)
59 \( 1 - 149.T + 2.05e5T^{2} \)
61 \( 1 + 495.T + 2.26e5T^{2} \)
67 \( 1 - 761. iT - 3.00e5T^{2} \)
71 \( 1 + 736.T + 3.57e5T^{2} \)
73 \( 1 - 701. iT - 3.89e5T^{2} \)
79 \( 1 + 780.T + 4.93e5T^{2} \)
83 \( 1 + 961. iT - 5.71e5T^{2} \)
89 \( 1 + 520.T + 7.04e5T^{2} \)
97 \( 1 + 1.15e3iT - 9.12e5T^{2} \)
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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

−10.58151967084093088324857363161, −9.985879606432729469838426892883, −8.705027200873575218034752352860, −7.42002105988849262698576564783, −7.00929809290229090458786345748, −5.72817864903850721060195028761, −4.20989385669906002668191742996, −3.15950862739873095931314630567, −2.22834449098329704560024888079, −0.44071984691028318746607449262, 1.48865560057287136862033443709, 3.08232850606519903236650166752, 4.48201003263983115533410231676, 5.52625496461409715729944374690, 6.45237695547853080958521820869, 7.66744324299038605037518903819, 8.076196626719727775008208073123, 9.103444034073529434996389319936, 10.34402325981895555734306693017, 11.29295744383849161425658048014

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