Properties

Label 2-405-1.1-c1-0-4
Degree $2$
Conductor $405$
Sign $1$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 5-s + 2·7-s + 3·11-s − 4·13-s + 4·16-s + 6·17-s − 19-s − 2·20-s + 6·23-s + 25-s − 4·28-s + 9·29-s − 31-s + 2·35-s + 8·37-s − 3·41-s − 4·43-s − 6·44-s − 12·47-s − 3·49-s + 8·52-s − 6·53-s + 3·55-s − 3·59-s − 10·61-s − 8·64-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s + 0.755·7-s + 0.904·11-s − 1.10·13-s + 16-s + 1.45·17-s − 0.229·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.755·28-s + 1.67·29-s − 0.179·31-s + 0.338·35-s + 1.31·37-s − 0.468·41-s − 0.609·43-s − 0.904·44-s − 1.75·47-s − 3/7·49-s + 1.10·52-s − 0.824·53-s + 0.404·55-s − 0.390·59-s − 1.28·61-s − 64-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $1$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{405} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.338654780\)
\(L(\frac12)\) \(\approx\) \(1.338654780\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good2 \( 1 + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−11.29810456021436859378746330171, −10.04624014100845215666127682432, −9.562964346295115144850959285425, −8.538687880420415041094117740367, −7.73102144096255023104365276788, −6.47289908140934830552102870516, −5.18690236062207152876422818521, −4.59634975877424853850281958730, −3.14233849758258863442381605357, −1.27645214139714047950709921081, 1.27645214139714047950709921081, 3.14233849758258863442381605357, 4.59634975877424853850281958730, 5.18690236062207152876422818521, 6.47289908140934830552102870516, 7.73102144096255023104365276788, 8.538687880420415041094117740367, 9.562964346295115144850959285425, 10.04624014100845215666127682432, 11.29810456021436859378746330171

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