Properties

Label 2-405-9.4-c3-0-8
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  + (−2.5 − 4.33i)2-s + (−8.50 + 14.7i)4-s + (2.5 − 4.33i)5-s + (15 + 25.9i)7-s + 45.0·8-s − 25.0·10-s + (−25 − 43.3i)11-s + (10 − 17.3i)13-s + (75.0 − 129. i)14-s + (−44.5 − 77.0i)16-s − 10·17-s − 44·19-s + (42.5 + 73.6i)20-s + (−125 + 216. i)22-s + (−60 + 103. i)23-s + ⋯
L(s)  = 1  + (−0.883 − 1.53i)2-s + (−1.06 + 1.84i)4-s + (0.223 − 0.387i)5-s + (0.809 + 1.40i)7-s + 1.98·8-s − 0.790·10-s + (−0.685 − 1.18i)11-s + (0.213 − 0.369i)13-s + (1.43 − 2.47i)14-s + (−0.695 − 1.20i)16-s − 0.142·17-s − 0.531·19-s + (0.475 + 0.823i)20-s + (−1.21 + 2.09i)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

\(L(2)\) \(\approx\) \(0.4719820540\)
\(L(\frac12)\) \(\approx\) \(0.4719820540\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.5 + 4.33i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-15 - 25.9i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (25 + 43.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-10 + 17.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 10T + 4.91e3T^{2} \)
19 \( 1 + 44T + 6.85e3T^{2} \)
23 \( 1 + (60 - 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-25 - 43.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (54 - 93.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 40T + 5.06e4T^{2} \)
41 \( 1 + (200 - 346. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (140 + 242. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-140 - 242. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 610T + 1.48e5T^{2} \)
59 \( 1 + (25 - 43.3i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-259 - 448. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-90 + 155. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 700T + 3.57e5T^{2} \)
73 \( 1 + 410T + 3.89e5T^{2} \)
79 \( 1 + (-258 - 446. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (330 + 571. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.50e3T + 7.04e5T^{2} \)
97 \( 1 + (-815 - 1.41e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.04271737016092636847719941447, −10.12289939200531715862184050419, −9.100788333211561553882095945943, −8.494686268336324330294094671012, −7.961726012138902725451146265600, −5.93029534567754306602593153570, −4.96900613364647962262309670031, −3.40066265727424851096887774839, −2.39407947475191098804945421754, −1.34940713884057245615358860871, 0.22064764322540520113011834597, 1.83196539973514570274446355534, 4.22877128469050909203900850297, 5.07638512658995306850145762071, 6.40565948983472398334174893231, 7.11891491713818793191777992537, 7.78274463408238423464839885345, 8.571891976392399784184266441378, 9.842876180679034293157990146802, 10.30666160217766735405054743280

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