Properties

Label 2-405-1.1-c3-0-36
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.31·2-s + 20.2·4-s + 5·5-s − 13.4·7-s + 65.1·8-s + 26.5·10-s + 46.9·11-s − 36.1·13-s − 71.4·14-s + 184.·16-s + 54.6·17-s + 111.·19-s + 101.·20-s + 249.·22-s + 35.9·23-s + 25·25-s − 192.·26-s − 272.·28-s − 58.1·29-s − 295.·31-s + 458.·32-s + 290.·34-s − 67.1·35-s − 53.0·37-s + 591.·38-s + 325.·40-s − 128.·41-s + ⋯
L(s)  = 1  + 1.87·2-s + 2.53·4-s + 0.447·5-s − 0.725·7-s + 2.87·8-s + 0.840·10-s + 1.28·11-s − 0.770·13-s − 1.36·14-s + 2.87·16-s + 0.779·17-s + 1.34·19-s + 1.13·20-s + 2.41·22-s + 0.326·23-s + 0.200·25-s − 1.44·26-s − 1.83·28-s − 0.372·29-s − 1.71·31-s + 2.53·32-s + 1.46·34-s − 0.324·35-s − 0.235·37-s + 2.52·38-s + 1.28·40-s − 0.488·41-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

\(L(2)\) \(\approx\) \(6.686789664\)
\(L(\frac12)\) \(\approx\) \(6.686789664\)
\(L(\frac{5}{2})\) 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 \( 1 - 5.31T + 8T^{2} \)
7 \( 1 + 13.4T + 343T^{2} \)
11 \( 1 - 46.9T + 1.33e3T^{2} \)
13 \( 1 + 36.1T + 2.19e3T^{2} \)
17 \( 1 - 54.6T + 4.91e3T^{2} \)
19 \( 1 - 111.T + 6.85e3T^{2} \)
23 \( 1 - 35.9T + 1.21e4T^{2} \)
29 \( 1 + 58.1T + 2.43e4T^{2} \)
31 \( 1 + 295.T + 2.97e4T^{2} \)
37 \( 1 + 53.0T + 5.06e4T^{2} \)
41 \( 1 + 128.T + 6.89e4T^{2} \)
43 \( 1 + 164.T + 7.95e4T^{2} \)
47 \( 1 + 87.8T + 1.03e5T^{2} \)
53 \( 1 - 479.T + 1.48e5T^{2} \)
59 \( 1 + 635.T + 2.05e5T^{2} \)
61 \( 1 - 48.0T + 2.26e5T^{2} \)
67 \( 1 + 28.9T + 3.00e5T^{2} \)
71 \( 1 + 576.T + 3.57e5T^{2} \)
73 \( 1 - 835.T + 3.89e5T^{2} \)
79 \( 1 + 203.T + 4.93e5T^{2} \)
83 \( 1 + 464.T + 5.71e5T^{2} \)
89 \( 1 + 993.T + 7.04e5T^{2} \)
97 \( 1 - 881.T + 9.12e5T^{2} \)
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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

−11.28760546133583254945866307823, −10.08856374234573974621487812824, −9.255161017167824126451993280697, −7.43595725733957509561741454221, −6.76999129326345828796136667438, −5.80471288431238686205647835172, −5.06044731752337269234862956210, −3.77669688988645145085316105998, −3.04068617154193777979897618394, −1.60773687503104671799500724727, 1.60773687503104671799500724727, 3.04068617154193777979897618394, 3.77669688988645145085316105998, 5.06044731752337269234862956210, 5.80471288431238686205647835172, 6.76999129326345828796136667438, 7.43595725733957509561741454221, 9.255161017167824126451993280697, 10.08856374234573974621487812824, 11.28760546133583254945866307823

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