Properties

Label 2-405-1.1-c3-0-8
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  − 3.04·2-s + 1.25·4-s + 5·5-s − 13.7·7-s + 20.5·8-s − 15.2·10-s − 31.8·11-s + 58.2·13-s + 41.7·14-s − 72.4·16-s − 109.·17-s + 129.·19-s + 6.26·20-s + 96.7·22-s − 79.6·23-s + 25·25-s − 177.·26-s − 17.1·28-s − 9.03·29-s − 33.3·31-s + 56.1·32-s + 331.·34-s − 68.5·35-s − 22.1·37-s − 394.·38-s + 102.·40-s − 121.·41-s + ⋯
L(s)  = 1  − 1.07·2-s + 0.156·4-s + 0.447·5-s − 0.740·7-s + 0.907·8-s − 0.480·10-s − 0.871·11-s + 1.24·13-s + 0.796·14-s − 1.13·16-s − 1.55·17-s + 1.56·19-s + 0.0699·20-s + 0.937·22-s − 0.722·23-s + 0.200·25-s − 1.33·26-s − 0.115·28-s − 0.0578·29-s − 0.193·31-s + 0.310·32-s + 1.67·34-s − 0.331·35-s − 0.0984·37-s − 1.68·38-s + 0.405·40-s − 0.463·41-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

\(L(2)\) \(\approx\) \(0.8236789412\)
\(L(\frac12)\) \(\approx\) \(0.8236789412\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.04T + 8T^{2} \)
7 \( 1 + 13.7T + 343T^{2} \)
11 \( 1 + 31.8T + 1.33e3T^{2} \)
13 \( 1 - 58.2T + 2.19e3T^{2} \)
17 \( 1 + 109.T + 4.91e3T^{2} \)
19 \( 1 - 129.T + 6.85e3T^{2} \)
23 \( 1 + 79.6T + 1.21e4T^{2} \)
29 \( 1 + 9.03T + 2.43e4T^{2} \)
31 \( 1 + 33.3T + 2.97e4T^{2} \)
37 \( 1 + 22.1T + 5.06e4T^{2} \)
41 \( 1 + 121.T + 6.89e4T^{2} \)
43 \( 1 - 10.1T + 7.95e4T^{2} \)
47 \( 1 - 441.T + 1.03e5T^{2} \)
53 \( 1 - 593.T + 1.48e5T^{2} \)
59 \( 1 + 442.T + 2.05e5T^{2} \)
61 \( 1 - 144.T + 2.26e5T^{2} \)
67 \( 1 - 862.T + 3.00e5T^{2} \)
71 \( 1 - 818.T + 3.57e5T^{2} \)
73 \( 1 - 495.T + 3.89e5T^{2} \)
79 \( 1 - 1.17e3T + 4.93e5T^{2} \)
83 \( 1 - 424.T + 5.71e5T^{2} \)
89 \( 1 + 1.03e3T + 7.04e5T^{2} \)
97 \( 1 - 1.59e3T + 9.12e5T^{2} \)
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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

−10.59815513951850156284932056849, −9.796628234948419242606762782976, −9.079965351346157717453494304303, −8.296712280698144675391933419983, −7.27635619845111808276828017670, −6.26495524476271136216442561222, −5.10888145568238799067577258423, −3.69799060440442134815543456491, −2.17885220313087044144120637342, −0.69071239505713741566432729649, 0.69071239505713741566432729649, 2.17885220313087044144120637342, 3.69799060440442134815543456491, 5.10888145568238799067577258423, 6.26495524476271136216442561222, 7.27635619845111808276828017670, 8.296712280698144675391933419983, 9.079965351346157717453494304303, 9.796628234948419242606762782976, 10.59815513951850156284932056849

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