Properties

Degree $2$
Conductor $4410$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s + 7·13-s + 16-s + 4·17-s + 19-s − 20-s − 22-s − 23-s + 25-s − 7·26-s + 8·29-s + 6·31-s − 32-s − 4·34-s − 3·37-s − 38-s + 40-s − 9·41-s − 4·43-s + 44-s + 46-s + 3·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s + 1.94·13-s + 1/4·16-s + 0.970·17-s + 0.229·19-s − 0.223·20-s − 0.213·22-s − 0.208·23-s + 1/5·25-s − 1.37·26-s + 1.48·29-s + 1.07·31-s − 0.176·32-s − 0.685·34-s − 0.493·37-s − 0.162·38-s + 0.158·40-s − 1.40·41-s − 0.609·43-s + 0.150·44-s + 0.147·46-s + 0.437·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4410\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4410} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4410,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.516429299\)
\(L(\frac12)\) \(\approx\) \(1.516429299\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good11 \( 1 - T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−18.11254815352074, −17.30624586850103, −16.77549778360071, −16.11650386974720, −15.59029467397230, −15.24018024736101, −14.02240424277275, −13.92119437749477, −12.92924792286896, −12.09926861266055, −11.72214557104416, −10.98725727437645, −10.35982328240127, −9.797230096043034, −8.845982359405383, −8.366843341319179, −7.907236327101362, −6.881305414882550, −6.363455893512125, −5.594025478224081, −4.552535434365263, −3.584890353479651, −3.025904503593833, −1.602325886671441, −0.8361249422280025, 0.8361249422280025, 1.602325886671441, 3.025904503593833, 3.584890353479651, 4.552535434365263, 5.594025478224081, 6.363455893512125, 6.881305414882550, 7.907236327101362, 8.366843341319179, 8.845982359405383, 9.797230096043034, 10.35982328240127, 10.98725727437645, 11.72214557104416, 12.09926861266055, 12.92924792286896, 13.92119437749477, 14.02240424277275, 15.24018024736101, 15.59029467397230, 16.11650386974720, 16.77549778360071, 17.30624586850103, 18.11254815352074

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