Properties

Label 2-405-1.1-c3-0-13
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  − 4.26·2-s + 10.1·4-s − 5·5-s + 30.7·7-s − 9.21·8-s + 21.3·10-s + 40.7·11-s + 63.2·13-s − 131.·14-s − 42.0·16-s − 6.58·17-s + 75.3·19-s − 50.8·20-s − 173.·22-s − 62.3·23-s + 25·25-s − 269.·26-s + 312.·28-s + 49.6·29-s + 103.·31-s + 252.·32-s + 28.0·34-s − 153.·35-s − 282.·37-s − 321.·38-s + 46.0·40-s − 157.·41-s + ⋯
L(s)  = 1  − 1.50·2-s + 1.27·4-s − 0.447·5-s + 1.66·7-s − 0.407·8-s + 0.673·10-s + 1.11·11-s + 1.34·13-s − 2.50·14-s − 0.656·16-s − 0.0940·17-s + 0.910·19-s − 0.568·20-s − 1.68·22-s − 0.565·23-s + 0.200·25-s − 2.03·26-s + 2.11·28-s + 0.317·29-s + 0.596·31-s + 1.39·32-s + 0.141·34-s − 0.742·35-s − 1.25·37-s − 1.37·38-s + 0.182·40-s − 0.600·41-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

\(L(2)\) \(\approx\) \(1.167918062\)
\(L(\frac12)\) \(\approx\) \(1.167918062\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.26T + 8T^{2} \)
7 \( 1 - 30.7T + 343T^{2} \)
11 \( 1 - 40.7T + 1.33e3T^{2} \)
13 \( 1 - 63.2T + 2.19e3T^{2} \)
17 \( 1 + 6.58T + 4.91e3T^{2} \)
19 \( 1 - 75.3T + 6.85e3T^{2} \)
23 \( 1 + 62.3T + 1.21e4T^{2} \)
29 \( 1 - 49.6T + 2.43e4T^{2} \)
31 \( 1 - 103.T + 2.97e4T^{2} \)
37 \( 1 + 282.T + 5.06e4T^{2} \)
41 \( 1 + 157.T + 6.89e4T^{2} \)
43 \( 1 + 337.T + 7.95e4T^{2} \)
47 \( 1 - 44.5T + 1.03e5T^{2} \)
53 \( 1 - 26.2T + 1.48e5T^{2} \)
59 \( 1 + 425.T + 2.05e5T^{2} \)
61 \( 1 - 850.T + 2.26e5T^{2} \)
67 \( 1 - 96.3T + 3.00e5T^{2} \)
71 \( 1 - 952.T + 3.57e5T^{2} \)
73 \( 1 + 50.8T + 3.89e5T^{2} \)
79 \( 1 + 197.T + 4.93e5T^{2} \)
83 \( 1 + 197.T + 5.71e5T^{2} \)
89 \( 1 - 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + 1.43e3T + 9.12e5T^{2} \)
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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

−10.81939904212213424802028127588, −9.842522722574301190127635484123, −8.658262277153445364084444410767, −8.407853250042967559280748385751, −7.47396556948554538029086414220, −6.48222691434272075392756449813, −4.97409491770830216879436917775, −3.76939229706880367161895096057, −1.77040465006059863744687634367, −0.984246551985948356398853228608, 0.984246551985948356398853228608, 1.77040465006059863744687634367, 3.76939229706880367161895096057, 4.97409491770830216879436917775, 6.48222691434272075392756449813, 7.47396556948554538029086414220, 8.407853250042967559280748385751, 8.658262277153445364084444410767, 9.842522722574301190127635484123, 10.81939904212213424802028127588

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