Properties

Label 2-405-1.1-c1-0-12
Degree $2$
Conductor $405$
Sign $1$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.51·2-s + 4.32·4-s + 5-s − 0.514·7-s + 5.83·8-s + 2.51·10-s − 3.32·11-s − 1.32·13-s − 1.29·14-s + 6.02·16-s − 3.32·17-s − 1.32·19-s + 4.32·20-s − 8.34·22-s + 4.12·23-s + 25-s − 3.32·26-s − 2.22·28-s − 1.38·29-s + 8.73·31-s + 3.48·32-s − 8.34·34-s − 0.514·35-s + 0.292·37-s − 3.32·38-s + 5.83·40-s − 11.3·41-s + ⋯
L(s)  = 1  + 1.77·2-s + 2.16·4-s + 0.447·5-s − 0.194·7-s + 2.06·8-s + 0.795·10-s − 1.00·11-s − 0.366·13-s − 0.345·14-s + 1.50·16-s − 0.805·17-s − 0.303·19-s + 0.966·20-s − 1.78·22-s + 0.860·23-s + 0.200·25-s − 0.651·26-s − 0.419·28-s − 0.257·29-s + 1.56·31-s + 0.616·32-s − 1.43·34-s − 0.0869·35-s + 0.0481·37-s − 0.538·38-s + 0.922·40-s − 1.77·41-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: no
Arithmetic: yes
Character: $\chi_{405} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.617205165\)
\(L(\frac12)\) \(\approx\) \(3.617205165\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.51T + 2T^{2} \)
7 \( 1 + 0.514T + 7T^{2} \)
11 \( 1 + 3.32T + 11T^{2} \)
13 \( 1 + 1.32T + 13T^{2} \)
17 \( 1 + 3.32T + 17T^{2} \)
19 \( 1 + 1.32T + 19T^{2} \)
23 \( 1 - 4.12T + 23T^{2} \)
29 \( 1 + 1.38T + 29T^{2} \)
31 \( 1 - 8.73T + 31T^{2} \)
37 \( 1 - 0.292T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 + 4.86T + 47T^{2} \)
53 \( 1 + 5.02T + 53T^{2} \)
59 \( 1 + 5.02T + 59T^{2} \)
61 \( 1 - 7.34T + 61T^{2} \)
67 \( 1 - 9.44T + 67T^{2} \)
71 \( 1 - 8.99T + 71T^{2} \)
73 \( 1 - 6.05T + 73T^{2} \)
79 \( 1 + 8.05T + 79T^{2} \)
83 \( 1 - 1.54T + 83T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−11.43647010305502459602123780382, −10.71476474098541103720363091295, −9.699823814832895875174538969106, −8.281294856092912784424661004141, −7.03024273615491346463579841902, −6.28996164367483920774021968805, −5.23647463857818594601923063419, −4.55802444124845438928070684110, −3.17689899663705654627478208061, −2.23768245130265216385360926189, 2.23768245130265216385360926189, 3.17689899663705654627478208061, 4.55802444124845438928070684110, 5.23647463857818594601923063419, 6.28996164367483920774021968805, 7.03024273615491346463579841902, 8.281294856092912784424661004141, 9.699823814832895875174538969106, 10.71476474098541103720363091295, 11.43647010305502459602123780382

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