Properties

Label 2-2352-1.1-c3-0-32
Degree $2$
Conductor $2352$
Sign $1$
Analytic cond. $138.772$
Root an. cond. $11.7801$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 0.239·5-s + 9·9-s − 8.09·11-s − 43.7·13-s − 0.717·15-s + 60.8·17-s − 98.8·19-s + 213.·23-s − 124.·25-s + 27·27-s + 110.·29-s − 80.0·31-s − 24.2·33-s − 2.88·37-s − 131.·39-s + 242.·41-s − 367.·43-s − 2.15·45-s + 89.3·47-s + 182.·51-s + 11.4·53-s + 1.93·55-s − 296.·57-s + 400.·59-s − 480.·61-s + 10.4·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.0214·5-s + 0.333·9-s − 0.221·11-s − 0.933·13-s − 0.0123·15-s + 0.867·17-s − 1.19·19-s + 1.93·23-s − 0.999·25-s + 0.192·27-s + 0.706·29-s − 0.463·31-s − 0.128·33-s − 0.0127·37-s − 0.539·39-s + 0.922·41-s − 1.30·43-s − 0.00713·45-s + 0.277·47-s + 0.500·51-s + 0.0295·53-s + 0.00475·55-s − 0.689·57-s + 0.882·59-s − 1.00·61-s + 0.0199·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(138.772\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.507270421\)
\(L(\frac12)\) \(\approx\) \(2.507270421\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
7 \( 1 \)
good5 \( 1 + 0.239T + 125T^{2} \)
11 \( 1 + 8.09T + 1.33e3T^{2} \)
13 \( 1 + 43.7T + 2.19e3T^{2} \)
17 \( 1 - 60.8T + 4.91e3T^{2} \)
19 \( 1 + 98.8T + 6.85e3T^{2} \)
23 \( 1 - 213.T + 1.21e4T^{2} \)
29 \( 1 - 110.T + 2.43e4T^{2} \)
31 \( 1 + 80.0T + 2.97e4T^{2} \)
37 \( 1 + 2.88T + 5.06e4T^{2} \)
41 \( 1 - 242.T + 6.89e4T^{2} \)
43 \( 1 + 367.T + 7.95e4T^{2} \)
47 \( 1 - 89.3T + 1.03e5T^{2} \)
53 \( 1 - 11.4T + 1.48e5T^{2} \)
59 \( 1 - 400.T + 2.05e5T^{2} \)
61 \( 1 + 480.T + 2.26e5T^{2} \)
67 \( 1 - 240.T + 3.00e5T^{2} \)
71 \( 1 - 978.T + 3.57e5T^{2} \)
73 \( 1 - 622.T + 3.89e5T^{2} \)
79 \( 1 - 545.T + 4.93e5T^{2} \)
83 \( 1 + 845.T + 5.71e5T^{2} \)
89 \( 1 - 203.T + 7.04e5T^{2} \)
97 \( 1 - 1.19e3T + 9.12e5T^{2} \)
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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

−8.606074999419479649530342798201, −7.88736512753257525933454043120, −7.20446852447837886263918285206, −6.44153208861240013995473055769, −5.35532190578459440178629268053, −4.66625675393868281850187475152, −3.66084924426145258091158332356, −2.79408338859058747538033365355, −1.94258543196174633309268858082, −0.67478570353209734172375947942, 0.67478570353209734172375947942, 1.94258543196174633309268858082, 2.79408338859058747538033365355, 3.66084924426145258091158332356, 4.66625675393868281850187475152, 5.35532190578459440178629268053, 6.44153208861240013995473055769, 7.20446852447837886263918285206, 7.88736512753257525933454043120, 8.606074999419479649530342798201

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