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Properties

Label 2-405-5.4-c3-0-44
Degree $2$
Conductor $405$
Sign $0.950 + 0.311i$
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  − 0.473i·2-s + 7.77·4-s + (10.6 + 3.48i)5-s + 8.20i·7-s − 7.47i·8-s + (1.64 − 5.03i)10-s + 5.26·11-s − 77.0i·13-s + 3.88·14-s + 58.6·16-s − 88.9i·17-s + 91.7·19-s + (82.6 + 27.0i)20-s − 2.49i·22-s + 154. i·23-s + ⋯
L(s)  = 1  − 0.167i·2-s + 0.971·4-s + (0.950 + 0.311i)5-s + 0.443i·7-s − 0.330i·8-s + (0.0521 − 0.159i)10-s + 0.144·11-s − 1.64i·13-s + 0.0742·14-s + 0.916·16-s − 1.26i·17-s + 1.10·19-s + (0.923 + 0.302i)20-s − 0.0241i·22-s + 1.40i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.950 + 0.311i$
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: \((2,\ 405,\ (\ :3/2),\ 0.950 + 0.311i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.039655164\)
\(L(\frac12)\) \(\approx\) \(3.039655164\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.6 - 3.48i)T \)
good2 \( 1 + 0.473iT - 8T^{2} \)
7 \( 1 - 8.20iT - 343T^{2} \)
11 \( 1 - 5.26T + 1.33e3T^{2} \)
13 \( 1 + 77.0iT - 2.19e3T^{2} \)
17 \( 1 + 88.9iT - 4.91e3T^{2} \)
19 \( 1 - 91.7T + 6.85e3T^{2} \)
23 \( 1 - 154. iT - 1.21e4T^{2} \)
29 \( 1 + 168.T + 2.43e4T^{2} \)
31 \( 1 - 72.7T + 2.97e4T^{2} \)
37 \( 1 + 154. iT - 5.06e4T^{2} \)
41 \( 1 - 7.95T + 6.89e4T^{2} \)
43 \( 1 - 23.2iT - 7.95e4T^{2} \)
47 \( 1 - 301. iT - 1.03e5T^{2} \)
53 \( 1 - 344. iT - 1.48e5T^{2} \)
59 \( 1 + 251.T + 2.05e5T^{2} \)
61 \( 1 - 272.T + 2.26e5T^{2} \)
67 \( 1 - 835. iT - 3.00e5T^{2} \)
71 \( 1 - 351.T + 3.57e5T^{2} \)
73 \( 1 + 522. iT - 3.89e5T^{2} \)
79 \( 1 - 700.T + 4.93e5T^{2} \)
83 \( 1 + 241. iT - 5.71e5T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + 389. iT - 9.12e5T^{2} \)
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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

−10.83822148875055567882181987481, −9.892267612640193413615775613510, −9.277328764545408851467944434573, −7.74943229665326043363313539028, −7.11364563482748209186393852439, −5.79093185993341243204220334265, −5.41647160471143881889920818791, −3.27461182310855808865273692728, −2.54632862161480116565311234363, −1.17057477339381003850420241220, 1.37801906851033588730787138402, 2.31171252029310234808799925274, 3.89129880953298427909780673299, 5.22035930281727116108666867216, 6.36195801406100407305668483775, 6.83254100286100083510289891879, 8.073103159939286026803235528209, 9.107847246972705620053992533808, 10.04111851590452942586631329186, 10.82725132095922203276842123291

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