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Properties

Label 2-405-9.7-c3-0-36
Degree $2$
Conductor $405$
Sign $-0.173 + 0.984i$
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  + (−1.92 + 3.33i)2-s + (−3.41 − 5.91i)4-s + (−2.5 − 4.33i)5-s + (13.1 − 22.8i)7-s − 4.52·8-s + 19.2·10-s + (−2.53 + 4.39i)11-s + (41.3 + 71.6i)13-s + (50.6 + 87.7i)14-s + (36.0 − 62.3i)16-s − 52.5·17-s − 29.8·19-s + (−17.0 + 29.5i)20-s + (−9.77 − 16.9i)22-s + (−49.1 − 85.1i)23-s + ⋯
L(s)  = 1  + (−0.680 + 1.17i)2-s + (−0.426 − 0.738i)4-s + (−0.223 − 0.387i)5-s + (0.710 − 1.23i)7-s − 0.199·8-s + 0.608·10-s + (−0.0696 + 0.120i)11-s + (0.882 + 1.52i)13-s + (0.967 + 1.67i)14-s + (0.562 − 0.974i)16-s − 0.750·17-s − 0.360·19-s + (−0.190 + 0.330i)20-s + (−0.0947 − 0.164i)22-s + (−0.445 − 0.771i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1886775449\)
\(L(\frac12)\) \(\approx\) \(0.1886775449\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.92 - 3.33i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (-13.1 + 22.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (2.53 - 4.39i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-41.3 - 71.6i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 52.5T + 4.91e3T^{2} \)
19 \( 1 + 29.8T + 6.85e3T^{2} \)
23 \( 1 + (49.1 + 85.1i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (83.9 - 145. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (95.3 + 165. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 365.T + 5.06e4T^{2} \)
41 \( 1 + (-55.8 - 96.7i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-201. + 349. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (116. - 201. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 410.T + 1.48e5T^{2} \)
59 \( 1 + (76.0 + 131. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (266. - 460. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (306. + 531. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 413.T + 3.57e5T^{2} \)
73 \( 1 + 114.T + 3.89e5T^{2} \)
79 \( 1 + (39.5 - 68.5i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-713. + 1.23e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 450.T + 7.04e5T^{2} \)
97 \( 1 + (718. - 1.24e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−10.53374792123963889344636238891, −9.196870754275281697899646170176, −8.683453781958320431084646434086, −7.70149995121552044016373174835, −7.00132373239749792482344448118, −6.16764325273512833413077740870, −4.75200273175611130245320914290, −3.88115627934379598153354938566, −1.63142408273439866584626582874, −0.078929629374126512802630378324, 1.57057442950052193918192380665, 2.64101080853975867462113051617, 3.63933704864360864537066731710, 5.33704660931981702498790704968, 6.20449522508300497173204072960, 7.86258537903474178250814259755, 8.547111163330149691833281390051, 9.300170578164413655996418525700, 10.45969830307219044392373544631, 10.99620860140613857445228976189

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