Properties

Label 2-405-1.1-c3-0-12
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  + 1.57·2-s − 5.53·4-s − 5·5-s + 34.2·7-s − 21.2·8-s − 7.85·10-s − 26.8·11-s − 19.4·13-s + 53.7·14-s + 10.9·16-s + 29.1·17-s + 47.1·19-s + 27.6·20-s − 42.2·22-s + 113.·23-s + 25·25-s − 30.6·26-s − 189.·28-s + 81.2·29-s − 10.8·31-s + 187.·32-s + 45.7·34-s − 171.·35-s + 410.·37-s + 73.9·38-s + 106.·40-s − 443.·41-s + ⋯
L(s)  = 1  + 0.555·2-s − 0.691·4-s − 0.447·5-s + 1.84·7-s − 0.939·8-s − 0.248·10-s − 0.737·11-s − 0.415·13-s + 1.02·14-s + 0.170·16-s + 0.415·17-s + 0.568·19-s + 0.309·20-s − 0.409·22-s + 1.02·23-s + 0.200·25-s − 0.230·26-s − 1.27·28-s + 0.520·29-s − 0.0627·31-s + 1.03·32-s + 0.230·34-s − 0.826·35-s + 1.82·37-s + 0.315·38-s + 0.420·40-s − 1.68·41-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

\(L(2)\) \(\approx\) \(2.194216716\)
\(L(\frac12)\) \(\approx\) \(2.194216716\)
\(L(\frac{5}{2})\) 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 \( 1 - 1.57T + 8T^{2} \)
7 \( 1 - 34.2T + 343T^{2} \)
11 \( 1 + 26.8T + 1.33e3T^{2} \)
13 \( 1 + 19.4T + 2.19e3T^{2} \)
17 \( 1 - 29.1T + 4.91e3T^{2} \)
19 \( 1 - 47.1T + 6.85e3T^{2} \)
23 \( 1 - 113.T + 1.21e4T^{2} \)
29 \( 1 - 81.2T + 2.43e4T^{2} \)
31 \( 1 + 10.8T + 2.97e4T^{2} \)
37 \( 1 - 410.T + 5.06e4T^{2} \)
41 \( 1 + 443.T + 6.89e4T^{2} \)
43 \( 1 - 339.T + 7.95e4T^{2} \)
47 \( 1 - 236.T + 1.03e5T^{2} \)
53 \( 1 - 609.T + 1.48e5T^{2} \)
59 \( 1 - 15.5T + 2.05e5T^{2} \)
61 \( 1 - 61.0T + 2.26e5T^{2} \)
67 \( 1 - 216.T + 3.00e5T^{2} \)
71 \( 1 - 65.4T + 3.57e5T^{2} \)
73 \( 1 - 711.T + 3.89e5T^{2} \)
79 \( 1 + 957.T + 4.93e5T^{2} \)
83 \( 1 + 522.T + 5.71e5T^{2} \)
89 \( 1 + 1.60e3T + 7.04e5T^{2} \)
97 \( 1 - 801.T + 9.12e5T^{2} \)
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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

−11.05107129385211003467575620361, −9.987023462173171780453155600818, −8.792799244282510171176000821966, −8.087637699293025828699597807943, −7.29971375068235128540608158173, −5.55567766833519892440067741919, −4.95335014449579350470096944370, −4.13546620660020685180015423952, −2.69859825181607269717792927407, −0.935079438675159345548818508351, 0.935079438675159345548818508351, 2.69859825181607269717792927407, 4.13546620660020685180015423952, 4.95335014449579350470096944370, 5.55567766833519892440067741919, 7.29971375068235128540608158173, 8.087637699293025828699597807943, 8.792799244282510171176000821966, 9.987023462173171780453155600818, 11.05107129385211003467575620361

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