Properties

Label 2-405-1.1-c3-0-18
Degree $2$
Conductor $405$
Sign $1$
Analytic cond. $23.8957$
Root an. cond. $4.88833$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.19·2-s − 3.18·4-s + 5·5-s + 2.76·7-s − 24.5·8-s + 10.9·10-s + 52.6·11-s + 20.4·13-s + 6.06·14-s − 28.3·16-s − 3.66·17-s − 95.6·19-s − 15.9·20-s + 115.·22-s + 89.8·23-s + 25·25-s + 44.8·26-s − 8.80·28-s + 227.·29-s + 279.·31-s + 134.·32-s − 8.03·34-s + 13.8·35-s + 273.·37-s − 209.·38-s − 122.·40-s − 64.8·41-s + ⋯
L(s)  = 1  + 0.775·2-s − 0.398·4-s + 0.447·5-s + 0.149·7-s − 1.08·8-s + 0.346·10-s + 1.44·11-s + 0.436·13-s + 0.115·14-s − 0.443·16-s − 0.0522·17-s − 1.15·19-s − 0.178·20-s + 1.11·22-s + 0.814·23-s + 0.200·25-s + 0.338·26-s − 0.0594·28-s + 1.45·29-s + 1.61·31-s + 0.740·32-s − 0.0405·34-s + 0.0667·35-s + 1.21·37-s − 0.896·38-s − 0.485·40-s − 0.247·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $1$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.830012147\)
\(L(\frac12)\) \(\approx\) \(2.830012147\)
\(L(\frac{5}{2})\) 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 \( 1 - 5T \)
good2 \( 1 - 2.19T + 8T^{2} \)
7 \( 1 - 2.76T + 343T^{2} \)
11 \( 1 - 52.6T + 1.33e3T^{2} \)
13 \( 1 - 20.4T + 2.19e3T^{2} \)
17 \( 1 + 3.66T + 4.91e3T^{2} \)
19 \( 1 + 95.6T + 6.85e3T^{2} \)
23 \( 1 - 89.8T + 1.21e4T^{2} \)
29 \( 1 - 227.T + 2.43e4T^{2} \)
31 \( 1 - 279.T + 2.97e4T^{2} \)
37 \( 1 - 273.T + 5.06e4T^{2} \)
41 \( 1 + 64.8T + 6.89e4T^{2} \)
43 \( 1 - 418.T + 7.95e4T^{2} \)
47 \( 1 + 138.T + 1.03e5T^{2} \)
53 \( 1 - 197.T + 1.48e5T^{2} \)
59 \( 1 + 741.T + 2.05e5T^{2} \)
61 \( 1 - 488.T + 2.26e5T^{2} \)
67 \( 1 + 411.T + 3.00e5T^{2} \)
71 \( 1 - 310.T + 3.57e5T^{2} \)
73 \( 1 + 51.0T + 3.89e5T^{2} \)
79 \( 1 - 1.20e3T + 4.93e5T^{2} \)
83 \( 1 + 905.T + 5.71e5T^{2} \)
89 \( 1 + 663.T + 7.04e5T^{2} \)
97 \( 1 + 725.T + 9.12e5T^{2} \)
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

−10.98680468478441212264617785909, −9.823774917532282220868394819548, −9.009356326545994669929452250141, −8.280438284161399266159875539124, −6.58502333747563300855318437903, −6.12540887557346043583908777185, −4.77479722836887099753179412606, −4.07298826479745293117054001121, −2.77856551342686772519634712802, −1.05306787406970586401934536607, 1.05306787406970586401934536607, 2.77856551342686772519634712802, 4.07298826479745293117054001121, 4.77479722836887099753179412606, 6.12540887557346043583908777185, 6.58502333747563300855318437903, 8.280438284161399266159875539124, 9.009356326545994669929452250141, 9.823774917532282220868394819548, 10.98680468478441212264617785909

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