Properties

Label 2-405-1.1-c3-0-32
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.52·2-s + 4.41·4-s + 5·5-s + 25.4·7-s + 12.6·8-s − 17.6·10-s − 71.3·11-s − 51.3·13-s − 89.6·14-s − 79.8·16-s − 33.3·17-s + 113.·19-s + 22.0·20-s + 251.·22-s + 81.9·23-s + 25·25-s + 181.·26-s + 112.·28-s − 246.·29-s + 222.·31-s + 180.·32-s + 117.·34-s + 127.·35-s + 22.3·37-s − 399.·38-s + 63.1·40-s − 434.·41-s + ⋯
L(s)  = 1  − 1.24·2-s + 0.551·4-s + 0.447·5-s + 1.37·7-s + 0.558·8-s − 0.557·10-s − 1.95·11-s − 1.09·13-s − 1.71·14-s − 1.24·16-s − 0.475·17-s + 1.36·19-s + 0.246·20-s + 2.43·22-s + 0.743·23-s + 0.200·25-s + 1.36·26-s + 0.757·28-s − 1.58·29-s + 1.29·31-s + 0.995·32-s + 0.592·34-s + 0.614·35-s + 0.0994·37-s − 1.70·38-s + 0.249·40-s − 1.65·41-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: \(1\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 3.52T + 8T^{2} \)
7 \( 1 - 25.4T + 343T^{2} \)
11 \( 1 + 71.3T + 1.33e3T^{2} \)
13 \( 1 + 51.3T + 2.19e3T^{2} \)
17 \( 1 + 33.3T + 4.91e3T^{2} \)
19 \( 1 - 113.T + 6.85e3T^{2} \)
23 \( 1 - 81.9T + 1.21e4T^{2} \)
29 \( 1 + 246.T + 2.43e4T^{2} \)
31 \( 1 - 222.T + 2.97e4T^{2} \)
37 \( 1 - 22.3T + 5.06e4T^{2} \)
41 \( 1 + 434.T + 6.89e4T^{2} \)
43 \( 1 + 236.T + 7.95e4T^{2} \)
47 \( 1 + 107.T + 1.03e5T^{2} \)
53 \( 1 + 123.T + 1.48e5T^{2} \)
59 \( 1 - 171.T + 2.05e5T^{2} \)
61 \( 1 + 79.4T + 2.26e5T^{2} \)
67 \( 1 + 611.T + 3.00e5T^{2} \)
71 \( 1 - 511.T + 3.57e5T^{2} \)
73 \( 1 + 410.T + 3.89e5T^{2} \)
79 \( 1 + 793.T + 4.93e5T^{2} \)
83 \( 1 - 270.T + 5.71e5T^{2} \)
89 \( 1 - 177.T + 7.04e5T^{2} \)
97 \( 1 + 881.T + 9.12e5T^{2} \)
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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

−10.22470468631406134068744240364, −9.551803346719183138579553726493, −8.444924049108022118121680814566, −7.79613708319798334569597549043, −7.14811395945593464516058885345, −5.25270629881691821845581184062, −4.86228557131038574859440496527, −2.64268932029356649570775336587, −1.54718867321210697574949638969, 0, 1.54718867321210697574949638969, 2.64268932029356649570775336587, 4.86228557131038574859440496527, 5.25270629881691821845581184062, 7.14811395945593464516058885345, 7.79613708319798334569597549043, 8.444924049108022118121680814566, 9.551803346719183138579553726493, 10.22470468631406134068744240364

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