Properties

Label 2-5520-1.1-c1-0-39
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 + 5·7-s + 9-s + 4·13-s − 15-s − 3·17-s + 4·19-s + 5·21-s + 23-s + 25-s + 27-s + 29-s − 31-s − 5·35-s − 37-s + 4·39-s + 11·41-s − 4·43-s − 45-s − 6·47-s + 18·49-s − 3·51-s + 53-s + 4·57-s + 59-s + 6·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1.88·7-s + 1/3·9-s + 1.10·13-s − 0.258·15-s − 0.727·17-s + 0.917·19-s + 1.09·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.185·29-s − 0.179·31-s − 0.845·35-s − 0.164·37-s + 0.640·39-s + 1.71·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s + 18/7·49-s − 0.420·51-s + 0.137·53-s + 0.529·57-s + 0.130·59-s + 0.768·61-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.276305109\)
\(L(\frac12)\) \(\approx\) \(3.276305109\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.244172834418588563827014645606, −7.58180751753748820471225455751, −6.98971243095208194554643329576, −5.92405361571973205732057203307, −5.11549302962715067142613094614, −4.42605182400092039828195277893, −3.80233378073640689226968908073, −2.79078782779929178015018546740, −1.79060270377778587290331546596, −1.03948134122832008320279790045, 1.03948134122832008320279790045, 1.79060270377778587290331546596, 2.79078782779929178015018546740, 3.80233378073640689226968908073, 4.42605182400092039828195277893, 5.11549302962715067142613094614, 5.92405361571973205732057203307, 6.98971243095208194554643329576, 7.58180751753748820471225455751, 8.244172834418588563827014645606

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