Properties

Label 2-405-1.1-c3-0-11
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  − 5.38·2-s + 20.9·4-s + 5·5-s + 12.5·7-s − 69.9·8-s − 26.9·10-s − 12.8·11-s + 59.8·13-s − 67.6·14-s + 208.·16-s + 110.·17-s − 12.0·19-s + 104.·20-s + 69.0·22-s + 67.7·23-s + 25·25-s − 322.·26-s + 263.·28-s − 199.·29-s + 76.6·31-s − 565.·32-s − 592.·34-s + 62.8·35-s − 22.4·37-s + 65.0·38-s − 349.·40-s − 87.7·41-s + ⋯
L(s)  = 1  − 1.90·2-s + 2.62·4-s + 0.447·5-s + 0.678·7-s − 3.09·8-s − 0.851·10-s − 0.351·11-s + 1.27·13-s − 1.29·14-s + 3.26·16-s + 1.56·17-s − 0.145·19-s + 1.17·20-s + 0.669·22-s + 0.613·23-s + 0.200·25-s − 2.43·26-s + 1.78·28-s − 1.28·29-s + 0.443·31-s − 3.12·32-s − 2.98·34-s + 0.303·35-s − 0.0998·37-s + 0.277·38-s − 1.38·40-s − 0.334·41-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

\(L(2)\) \(\approx\) \(1.025019232\)
\(L(\frac12)\) \(\approx\) \(1.025019232\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.38T + 8T^{2} \)
7 \( 1 - 12.5T + 343T^{2} \)
11 \( 1 + 12.8T + 1.33e3T^{2} \)
13 \( 1 - 59.8T + 2.19e3T^{2} \)
17 \( 1 - 110.T + 4.91e3T^{2} \)
19 \( 1 + 12.0T + 6.85e3T^{2} \)
23 \( 1 - 67.7T + 1.21e4T^{2} \)
29 \( 1 + 199.T + 2.43e4T^{2} \)
31 \( 1 - 76.6T + 2.97e4T^{2} \)
37 \( 1 + 22.4T + 5.06e4T^{2} \)
41 \( 1 + 87.7T + 6.89e4T^{2} \)
43 \( 1 - 119.T + 7.95e4T^{2} \)
47 \( 1 + 243.T + 1.03e5T^{2} \)
53 \( 1 - 293.T + 1.48e5T^{2} \)
59 \( 1 - 581.T + 2.05e5T^{2} \)
61 \( 1 + 773.T + 2.26e5T^{2} \)
67 \( 1 + 231.T + 3.00e5T^{2} \)
71 \( 1 - 744.T + 3.57e5T^{2} \)
73 \( 1 + 264.T + 3.89e5T^{2} \)
79 \( 1 - 559.T + 4.93e5T^{2} \)
83 \( 1 - 1.22e3T + 5.71e5T^{2} \)
89 \( 1 - 255.T + 7.04e5T^{2} \)
97 \( 1 - 1.04e3T + 9.12e5T^{2} \)
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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

−10.66063292353133290426179947900, −9.841671267809869281109140820217, −8.995512685602000406236403916230, −8.181490124276465165011747565119, −7.52603847878894309490684595416, −6.38579725004132591996967345325, −5.43833683451724437976185389796, −3.29910180872050006304963293809, −1.87431982926382245707977424244, −0.924802202868821414164549930099, 0.924802202868821414164549930099, 1.87431982926382245707977424244, 3.29910180872050006304963293809, 5.43833683451724437976185389796, 6.38579725004132591996967345325, 7.52603847878894309490684595416, 8.181490124276465165011747565119, 8.995512685602000406236403916230, 9.841671267809869281109140820217, 10.66063292353133290426179947900

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