Properties

Label 2-5520-1.1-c1-0-23
Degree $2$
Conductor $5520$
Sign $1$
Analytic cond. $44.0774$
Root an. cond. $6.63908$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 3.58·7-s + 9-s + 0.715·13-s − 15-s − 1.58·17-s − 3.58·21-s − 23-s + 25-s − 27-s + 5.58·29-s − 4.87·31-s + 3.58·35-s + 8.15·37-s − 0.715·39-s + 4.30·41-s − 8.45·43-s + 45-s + 7.17·47-s + 5.87·49-s + 1.58·51-s + 8.30·53-s + 2.30·59-s + 6.45·61-s + 3.58·63-s + 0.715·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.35·7-s + 0.333·9-s + 0.198·13-s − 0.258·15-s − 0.385·17-s − 0.782·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 1.03·29-s − 0.875·31-s + 0.606·35-s + 1.34·37-s − 0.114·39-s + 0.672·41-s − 1.29·43-s + 0.149·45-s + 1.04·47-s + 0.838·49-s + 0.222·51-s + 1.14·53-s + 0.299·59-s + 0.827·61-s + 0.452·63-s + 0.0887·65-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.192663113\)
\(L(\frac12)\) \(\approx\) \(2.192663113\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.58T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 0.715T + 13T^{2} \)
17 \( 1 + 1.58T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 - 5.58T + 29T^{2} \)
31 \( 1 + 4.87T + 31T^{2} \)
37 \( 1 - 8.15T + 37T^{2} \)
41 \( 1 - 4.30T + 41T^{2} \)
43 \( 1 + 8.45T + 43T^{2} \)
47 \( 1 - 7.17T + 47T^{2} \)
53 \( 1 - 8.30T + 53T^{2} \)
59 \( 1 - 2.30T + 59T^{2} \)
61 \( 1 - 6.45T + 61T^{2} \)
67 \( 1 - 6.15T + 67T^{2} \)
71 \( 1 - 1.01T + 71T^{2} \)
73 \( 1 + 5.17T + 73T^{2} \)
79 \( 1 + 9.89T + 79T^{2} \)
83 \( 1 + 5.01T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + 6.45T + 97T^{2} \)
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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

−8.177738195546493942115476530995, −7.42031240882041207671308055881, −6.69974539443864719530235085948, −5.89315056971853378827967095317, −5.28998146582063559999995082271, −4.59952912990860456750447105199, −3.92862823202806803796641588463, −2.61220570420147020495200672233, −1.78248115449682135411049659237, −0.858417924438242364045258630289, 0.858417924438242364045258630289, 1.78248115449682135411049659237, 2.61220570420147020495200672233, 3.92862823202806803796641588463, 4.59952912990860456750447105199, 5.28998146582063559999995082271, 5.89315056971853378827967095317, 6.69974539443864719530235085948, 7.42031240882041207671308055881, 8.177738195546493942115476530995

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