↑

Properties

Label 2-405-15.14-c4-0-69
Degree $2$
Conductor $405$
Sign $-0.970 + 0.240i$
Analytic cond. $41.8648$
Root an. cond. $6.47030$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.633·2-s − 15.5·4-s + (−6.01 − 24.2i)5-s + 84.5i·7-s + 20.0·8-s + (3.80 + 15.3i)10-s − 171. i·11-s + 146. i·13-s − 53.5i·14-s + 236.·16-s + 273.·17-s + 229.·19-s + (93.8 + 378. i)20-s + 108. i·22-s − 389.·23-s + ⋯
L(s)  = 1  − 0.158·2-s − 0.974·4-s + (−0.240 − 0.970i)5-s + 1.72i·7-s + 0.312·8-s + (0.0380 + 0.153i)10-s − 1.41i·11-s + 0.869i·13-s − 0.273i·14-s + 0.925·16-s + 0.946·17-s + 0.636·19-s + (0.234 + 0.946i)20-s + 0.224i·22-s − 0.735·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $-0.970 + 0.240i$
Analytic conductor: \(41.8648\)
Root analytic conductor: \(6.47030\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :2),\ -0.970 + 0.240i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1695575829\)
\(L(\frac12)\) \(\approx\) \(0.1695575829\)
\(L(3)\) 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 \( 1 + (6.01 + 24.2i)T \)
good2 \( 1 + 0.633T + 16T^{2} \)
7 \( 1 - 84.5iT - 2.40e3T^{2} \)
11 \( 1 + 171. iT - 1.46e4T^{2} \)
13 \( 1 - 146. iT - 2.85e4T^{2} \)
17 \( 1 - 273.T + 8.35e4T^{2} \)
19 \( 1 - 229.T + 1.30e5T^{2} \)
23 \( 1 + 389.T + 2.79e5T^{2} \)
29 \( 1 + 354. iT - 7.07e5T^{2} \)
31 \( 1 - 621.T + 9.23e5T^{2} \)
37 \( 1 + 730. iT - 1.87e6T^{2} \)
41 \( 1 + 576. iT - 2.82e6T^{2} \)
43 \( 1 - 904. iT - 3.41e6T^{2} \)
47 \( 1 + 792.T + 4.87e6T^{2} \)
53 \( 1 + 3.46e3T + 7.89e6T^{2} \)
59 \( 1 - 2.81e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.32e3T + 1.38e7T^{2} \)
67 \( 1 - 170. iT - 2.01e7T^{2} \)
71 \( 1 + 2.79e3iT - 2.54e7T^{2} \)
73 \( 1 - 5.86e3iT - 2.83e7T^{2} \)
79 \( 1 + 2.93e3T + 3.89e7T^{2} \)
83 \( 1 + 6.88e3T + 4.74e7T^{2} \)
89 \( 1 - 1.03e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.40e4iT - 8.85e7T^{2} \)
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

−9.807953707248002743445426066803, −9.155619876683284199350996286155, −8.521904144393002033950364717975, −7.913681922620401505720461417580, −5.94729816944983780137893188037, −5.46777164369718663775971182303, −4.37053114919840135544555295409, −3.11642883427326715797453920128, −1.41682156255081233646318143862, −0.05965222587608519450225881225, 1.25439917760327778396223587288, 3.22813499541013654409709521180, 4.08803418903539043189447566749, 5.04485034262577184224562152114, 6.55394395634162954934333786253, 7.68045504662630700598092744001, 7.81783376910138824289090078111, 9.612911881503467814848476282639, 10.15356760859943829609765500258, 10.63956236894553580279513125349

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