Properties

Label 2-405-135.83-c1-0-7
Degree $2$
Conductor $405$
Sign $0.997 - 0.0729i$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 + 0.866i)2-s + (0.0960 − 0.264i)4-s + (2.23 − 0.140i)5-s + (0.677 − 1.45i)7-s + (−0.671 − 2.50i)8-s + (−2.63 + 2.10i)10-s + (1.78 − 2.12i)11-s + (0.176 − 0.251i)13-s + (0.420 + 2.38i)14-s + (3.43 + 2.88i)16-s + (1.88 − 7.02i)17-s + (−5.07 − 2.92i)19-s + (0.177 − 0.602i)20-s + (−0.365 + 4.17i)22-s + (−0.116 + 0.0543i)23-s + ⋯
L(s)  = 1  + (−0.874 + 0.612i)2-s + (0.0480 − 0.132i)4-s + (0.998 − 0.0626i)5-s + (0.256 − 0.549i)7-s + (−0.237 − 0.886i)8-s + (−0.834 + 0.666i)10-s + (0.538 − 0.641i)11-s + (0.0489 − 0.0698i)13-s + (0.112 + 0.637i)14-s + (0.858 + 0.720i)16-s + (0.456 − 1.70i)17-s + (−1.16 − 0.671i)19-s + (0.0396 − 0.134i)20-s + (−0.0779 + 0.890i)22-s + (−0.0243 + 0.0113i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.997 - 0.0729i$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 0.997 - 0.0729i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01184 + 0.0369660i\)
\(L(\frac12)\) \(\approx\) \(1.01184 + 0.0369660i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.140i)T \)
good2 \( 1 + (1.23 - 0.866i)T + (0.684 - 1.87i)T^{2} \)
7 \( 1 + (-0.677 + 1.45i)T + (-4.49 - 5.36i)T^{2} \)
11 \( 1 + (-1.78 + 2.12i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-0.176 + 0.251i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (-1.88 + 7.02i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.07 + 2.92i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.116 - 0.0543i)T + (14.7 - 17.6i)T^{2} \)
29 \( 1 + (1.51 - 8.60i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-4.22 - 1.53i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-2.08 - 0.558i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.62 + 0.815i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (0.429 + 4.90i)T + (-42.3 + 7.46i)T^{2} \)
47 \( 1 + (-7.69 - 3.58i)T + (30.2 + 36.0i)T^{2} \)
53 \( 1 + (0.483 - 0.483i)T - 53iT^{2} \)
59 \( 1 + (7.43 - 6.23i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.50 + 0.910i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.884 + 0.619i)T + (22.9 + 62.9i)T^{2} \)
71 \( 1 + (9.03 - 5.21i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.65 - 1.78i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.04 - 0.360i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (7.55 + 10.7i)T + (-28.3 + 77.9i)T^{2} \)
89 \( 1 + (-5.58 + 9.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.8 - 1.03i)T + (95.5 - 16.8i)T^{2} \)
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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

−10.94058328076309911998894802906, −10.17135610283004046741001158902, −9.120571263233791659293045776500, −8.795933744966146792501165982043, −7.47600435812684001249750872067, −6.79844321737405234907189323156, −5.79205702960526189549181450036, −4.47930976027817639198192466237, −2.94399926529676460878386637414, −0.997468405866986239595126058438, 1.57589829850090026820791324647, 2.38002499542498979933942664669, 4.25868053698845498193951442147, 5.72312925631643775027008076996, 6.28654883550746059867467762382, 7.963998164760337788392798433220, 8.707624913431797494902189933301, 9.605861325904057947697840866392, 10.19551161472311390621711436043, 10.94978953432935195365099606306

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