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Properties

Label 2-405-5.4-c3-0-0
Degree $2$
Conductor $405$
Sign $0.851 + 0.524i$
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  − 5.49i·2-s − 22.1·4-s + (−9.52 − 5.85i)5-s − 19.7i·7-s + 78.0i·8-s + (−32.1 + 52.3i)10-s − 61.3·11-s + 7.73i·13-s − 108.·14-s + 251.·16-s − 42.7i·17-s − 19.5·19-s + (211. + 130. i)20-s + 337. i·22-s − 25.3i·23-s + ⋯
L(s)  = 1  − 1.94i·2-s − 2.77·4-s + (−0.851 − 0.524i)5-s − 1.06i·7-s + 3.44i·8-s + (−1.01 + 1.65i)10-s − 1.68·11-s + 0.165i·13-s − 2.06·14-s + 3.92·16-s − 0.609i·17-s − 0.236·19-s + (2.36 + 1.45i)20-s + 3.26i·22-s − 0.229i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.851 + 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.851 + 0.524i$
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: \((2,\ 405,\ (\ :3/2),\ 0.851 + 0.524i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.04700851599\)
\(L(\frac12)\) \(\approx\) \(0.04700851599\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.52 + 5.85i)T \)
good2 \( 1 + 5.49iT - 8T^{2} \)
7 \( 1 + 19.7iT - 343T^{2} \)
11 \( 1 + 61.3T + 1.33e3T^{2} \)
13 \( 1 - 7.73iT - 2.19e3T^{2} \)
17 \( 1 + 42.7iT - 4.91e3T^{2} \)
19 \( 1 + 19.5T + 6.85e3T^{2} \)
23 \( 1 + 25.3iT - 1.21e4T^{2} \)
29 \( 1 + 74.0T + 2.43e4T^{2} \)
31 \( 1 - 193.T + 2.97e4T^{2} \)
37 \( 1 + 273. iT - 5.06e4T^{2} \)
41 \( 1 + 91.6T + 6.89e4T^{2} \)
43 \( 1 - 142. iT - 7.95e4T^{2} \)
47 \( 1 + 548. iT - 1.03e5T^{2} \)
53 \( 1 - 385. iT - 1.48e5T^{2} \)
59 \( 1 - 228.T + 2.05e5T^{2} \)
61 \( 1 + 113.T + 2.26e5T^{2} \)
67 \( 1 + 286. iT - 3.00e5T^{2} \)
71 \( 1 + 763.T + 3.57e5T^{2} \)
73 \( 1 - 528. iT - 3.89e5T^{2} \)
79 \( 1 - 222.T + 4.93e5T^{2} \)
83 \( 1 + 484. iT - 5.71e5T^{2} \)
89 \( 1 - 1.31e3T + 7.04e5T^{2} \)
97 \( 1 - 813. iT - 9.12e5T^{2} \)
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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

−10.70530370813131343988781591472, −10.29889800697577361654282203397, −9.206430698145096012303300695169, −8.240802378341273305938882005956, −7.47974183041622929778768575617, −5.24497689904950172218887471323, −4.47570320918170497964965426364, −3.58192854669569031515939457257, −2.44028373027061457079508842751, −0.859281215124915818884805359555, 0.02237858316647969258805976306, 3.02169725129413186688167515526, 4.45073038569076917225167122750, 5.39068318920011308569016622516, 6.20357328046846584519531348407, 7.22417345780533236233578584110, 8.136571763685156822917118067882, 8.426529939751095169195995872768, 9.703645708270328329188610465239, 10.66517830123975899244656504497

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