Properties

Label 2-5520-1.1-c1-0-47
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 + 3.82·7-s + 9-s − 1.41·11-s + 3.41·13-s + 15-s + 0.414·17-s + 4.58·19-s + 3.82·21-s + 23-s + 25-s + 27-s − 6.41·29-s + 31-s − 1.41·33-s + 3.82·35-s + 9.48·37-s + 3.41·39-s − 1.24·41-s − 3.65·43-s + 45-s − 2.24·47-s + 7.65·49-s + 0.414·51-s − 3.24·53-s − 1.41·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.44·7-s + 0.333·9-s − 0.426·11-s + 0.946·13-s + 0.258·15-s + 0.100·17-s + 1.05·19-s + 0.835·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 1.19·29-s + 0.179·31-s − 0.246·33-s + 0.647·35-s + 1.55·37-s + 0.546·39-s − 0.194·41-s − 0.557·43-s + 0.149·45-s − 0.327·47-s + 1.09·49-s + 0.0580·51-s − 0.445·53-s − 0.190·55-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

\(L(1)\) \(\approx\) \(3.639019673\)
\(L(\frac12)\) \(\approx\) \(3.639019673\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.82T + 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 - 3.41T + 13T^{2} \)
17 \( 1 - 0.414T + 17T^{2} \)
19 \( 1 - 4.58T + 19T^{2} \)
29 \( 1 + 6.41T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 - 9.48T + 37T^{2} \)
41 \( 1 + 1.24T + 41T^{2} \)
43 \( 1 + 3.65T + 43T^{2} \)
47 \( 1 + 2.24T + 47T^{2} \)
53 \( 1 + 3.24T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 + 1.48T + 67T^{2} \)
71 \( 1 + 0.0710T + 71T^{2} \)
73 \( 1 + 6.24T + 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−8.104975851131140692663251958422, −7.67038220229458437581256168661, −6.86284105408132507350035227485, −5.84447752046096597333364242467, −5.27606114813668981785376092263, −4.51658444697396938229746598583, −3.66190094936129529957273659331, −2.74679469783100957452263129259, −1.82172341760027644092478262511, −1.10247808635851129708709328847, 1.10247808635851129708709328847, 1.82172341760027644092478262511, 2.74679469783100957452263129259, 3.66190094936129529957273659331, 4.51658444697396938229746598583, 5.27606114813668981785376092263, 5.84447752046096597333364242467, 6.86284105408132507350035227485, 7.67038220229458437581256168661, 8.104975851131140692663251958422

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