↑

Properties

Label 2-405-9.7-c3-0-3
Degree $2$
Conductor $405$
Sign $-0.984 - 0.173i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0745 − 0.129i)2-s + (3.98 + 6.90i)4-s + (−2.5 − 4.33i)5-s + (−10.0 + 17.4i)7-s + 2.38·8-s − 0.745·10-s + (4.94 − 8.56i)11-s + (5.91 + 10.2i)13-s + (1.50 + 2.60i)14-s + (−31.7 + 54.9i)16-s − 6.09·17-s − 62.6·19-s + (19.9 − 34.5i)20-s + (−0.737 − 1.27i)22-s + (−6.03 − 10.4i)23-s + ⋯
L(s)  = 1  + (0.0263 − 0.0456i)2-s + (0.498 + 0.863i)4-s + (−0.223 − 0.387i)5-s + (−0.543 + 0.941i)7-s + 0.105·8-s − 0.0235·10-s + (0.135 − 0.234i)11-s + (0.126 + 0.218i)13-s + (0.0286 + 0.0496i)14-s + (−0.495 + 0.858i)16-s − 0.0869·17-s − 0.756·19-s + (0.222 − 0.386i)20-s + (−0.00715 − 0.0123i)22-s + (−0.0547 − 0.0948i)23-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.984 - 0.173i$
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: \((2,\ 405,\ (\ :3/2),\ -0.984 - 0.173i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7524639395\)
\(L(\frac12)\) \(\approx\) \(0.7524639395\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.0745 + 0.129i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (10.0 - 17.4i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-4.94 + 8.56i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-5.91 - 10.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 6.09T + 4.91e3T^{2} \)
19 \( 1 + 62.6T + 6.85e3T^{2} \)
23 \( 1 + (6.03 + 10.4i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (70.2 - 121. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (89.0 + 154. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 216.T + 5.06e4T^{2} \)
41 \( 1 + (205. + 356. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-24.4 + 42.3i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (307. - 532. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 705.T + 1.48e5T^{2} \)
59 \( 1 + (247. + 428. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (333. - 576. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (138. + 240. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 239.T + 3.57e5T^{2} \)
73 \( 1 + 919.T + 3.89e5T^{2} \)
79 \( 1 + (258. - 447. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-326. + 564. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 543.T + 7.04e5T^{2} \)
97 \( 1 + (566. - 981. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.44411531999652265358622135013, −10.50390873862034427144511164960, −9.095737717040818678603562989523, −8.672527116775146071214070665566, −7.57650143003591304363973401078, −6.59141511143320833434399403735, −5.61413743725110546540307113979, −4.18974989863439991607422316099, −3.14723168362126079989478634634, −1.96538845222514004466506968513, 0.22872869658591736297526059245, 1.74049013693790037521381984840, 3.23536226095276999278676552951, 4.43813665817805918212228460868, 5.72527698575963147682412408426, 6.74220507134058874454138898782, 7.25060698728470186997872879554, 8.571690695493476164355087719660, 9.868250863127291951955726082836, 10.34155755584120573018282424621

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