Properties

Label 2-2752-1.1-c1-0-1
Degree $2$
Conductor $2752$
Sign $1$
Analytic cond. $21.9748$
Root an. cond. $4.68773$
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  − 0.185·3-s − 3.29·5-s − 2.67·7-s − 2.96·9-s − 3.10·11-s − 5.10·13-s + 0.612·15-s + 1.75·17-s − 3.96·19-s + 0.496·21-s − 1.23·23-s + 5.85·25-s + 1.10·27-s − 2.68·29-s − 0.123·31-s + 0.577·33-s + 8.79·35-s − 7.07·37-s + 0.948·39-s − 4.20·41-s − 43-s + 9.77·45-s + 1.14·47-s + 0.131·49-s − 0.325·51-s + 11.6·53-s + 10.2·55-s + ⋯
L(s)  = 1  − 0.107·3-s − 1.47·5-s − 1.00·7-s − 0.988·9-s − 0.936·11-s − 1.41·13-s + 0.158·15-s + 0.424·17-s − 0.909·19-s + 0.108·21-s − 0.257·23-s + 1.17·25-s + 0.213·27-s − 0.498·29-s − 0.0221·31-s + 0.100·33-s + 1.48·35-s − 1.16·37-s + 0.151·39-s − 0.656·41-s − 0.152·43-s + 1.45·45-s + 0.166·47-s + 0.0188·49-s − 0.0456·51-s + 1.60·53-s + 1.37·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2752\)    =    \(2^{6} \cdot 43\)
Sign: $1$
Analytic conductor: \(21.9748\)
Root analytic conductor: \(4.68773\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2752,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1142147827\)
\(L(\frac12)\) \(\approx\) \(0.1142147827\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 + T \)
good3 \( 1 + 0.185T + 3T^{2} \)
5 \( 1 + 3.29T + 5T^{2} \)
7 \( 1 + 2.67T + 7T^{2} \)
11 \( 1 + 3.10T + 11T^{2} \)
13 \( 1 + 5.10T + 13T^{2} \)
17 \( 1 - 1.75T + 17T^{2} \)
19 \( 1 + 3.96T + 19T^{2} \)
23 \( 1 + 1.23T + 23T^{2} \)
29 \( 1 + 2.68T + 29T^{2} \)
31 \( 1 + 0.123T + 31T^{2} \)
37 \( 1 + 7.07T + 37T^{2} \)
41 \( 1 + 4.20T + 41T^{2} \)
47 \( 1 - 1.14T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 12.9T + 61T^{2} \)
67 \( 1 - 0.143T + 67T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 - 7.63T + 73T^{2} \)
79 \( 1 - 16.2T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 1.16T + 89T^{2} \)
97 \( 1 + 7.53T + 97T^{2} \)
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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

−8.633140777775052025262533083774, −8.062766318848821225995115222625, −7.34521639217847201706228696789, −6.70933630636398340907198268747, −5.64227831834301405201736546438, −4.94245431886113051775949584363, −3.93268563759966208375648187189, −3.16846246315625988527506676078, −2.40002615500226152100262098378, −0.19249320238573546797170423513, 0.19249320238573546797170423513, 2.40002615500226152100262098378, 3.16846246315625988527506676078, 3.93268563759966208375648187189, 4.94245431886113051775949584363, 5.64227831834301405201736546438, 6.70933630636398340907198268747, 7.34521639217847201706228696789, 8.062766318848821225995115222625, 8.633140777775052025262533083774

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