Properties

Label 2-5520-1.1-c1-0-35
Degree $2$
Conductor $5520$
Sign $1$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
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  − 3-s + 5-s + 9-s + 2·11-s + 4·13-s − 15-s + 6·17-s + 8·19-s + 23-s + 25-s − 27-s + 4·29-s − 2·33-s − 2·37-s − 4·39-s − 2·41-s − 2·43-s + 45-s + 12·47-s − 7·49-s − 6·51-s − 6·53-s + 2·55-s − 8·57-s − 8·59-s + 2·61-s + 4·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 1.10·13-s − 0.258·15-s + 1.45·17-s + 1.83·19-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 0.348·33-s − 0.328·37-s − 0.640·39-s − 0.312·41-s − 0.304·43-s + 0.149·45-s + 1.75·47-s − 49-s − 0.840·51-s − 0.824·53-s + 0.269·55-s − 1.05·57-s − 1.04·59-s + 0.256·61-s + 0.496·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(44.0774\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.344413734\)
\(L(\frac12)\) \(\approx\) \(2.344413734\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
23 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
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

−8.099829289251043056017865920156, −7.36774795875050200479729686513, −6.66016768918830084650333494318, −5.87162779404798123328649467192, −5.46774177718589897563459500213, −4.61153870419235948642789565496, −3.58619045821314009673705835228, −3.00826336935468801990164382087, −1.51322780352927555501790305391, −0.972194799436486776940325096095, 0.972194799436486776940325096095, 1.51322780352927555501790305391, 3.00826336935468801990164382087, 3.58619045821314009673705835228, 4.61153870419235948642789565496, 5.46774177718589897563459500213, 5.87162779404798123328649467192, 6.66016768918830084650333494318, 7.36774795875050200479729686513, 8.099829289251043056017865920156

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