Properties

Label 2-2352-1.1-c3-0-14
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 − 12.4·5-s + 9·9-s − 60.3·11-s + 36.4·13-s − 37.3·15-s − 48.7·17-s + 50.5·19-s − 138.·23-s + 29.6·25-s + 27·27-s − 61.1·29-s + 1.16·31-s − 180.·33-s + 69.5·37-s + 109.·39-s + 308.·41-s − 174.·43-s − 111.·45-s − 389.·47-s − 146.·51-s + 314.·53-s + 749.·55-s + 151.·57-s − 844.·59-s − 338.·61-s − 452.·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.11·5-s + 0.333·9-s − 1.65·11-s + 0.777·13-s − 0.642·15-s − 0.695·17-s + 0.610·19-s − 1.25·23-s + 0.236·25-s + 0.192·27-s − 0.391·29-s + 0.00677·31-s − 0.954·33-s + 0.308·37-s + 0.448·39-s + 1.17·41-s − 0.618·43-s − 0.370·45-s − 1.20·47-s − 0.401·51-s + 0.816·53-s + 1.83·55-s + 0.352·57-s − 1.86·59-s − 0.710·61-s − 0.864·65-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: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.252321385\)
\(L(\frac12)\) \(\approx\) \(1.252321385\)
\(L(\frac{5}{2})\) 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 \( 1 + 12.4T + 125T^{2} \)
11 \( 1 + 60.3T + 1.33e3T^{2} \)
13 \( 1 - 36.4T + 2.19e3T^{2} \)
17 \( 1 + 48.7T + 4.91e3T^{2} \)
19 \( 1 - 50.5T + 6.85e3T^{2} \)
23 \( 1 + 138.T + 1.21e4T^{2} \)
29 \( 1 + 61.1T + 2.43e4T^{2} \)
31 \( 1 - 1.16T + 2.97e4T^{2} \)
37 \( 1 - 69.5T + 5.06e4T^{2} \)
41 \( 1 - 308.T + 6.89e4T^{2} \)
43 \( 1 + 174.T + 7.95e4T^{2} \)
47 \( 1 + 389.T + 1.03e5T^{2} \)
53 \( 1 - 314.T + 1.48e5T^{2} \)
59 \( 1 + 844.T + 2.05e5T^{2} \)
61 \( 1 + 338.T + 2.26e5T^{2} \)
67 \( 1 - 971.T + 3.00e5T^{2} \)
71 \( 1 - 98.4T + 3.57e5T^{2} \)
73 \( 1 - 710.T + 3.89e5T^{2} \)
79 \( 1 - 486.T + 4.93e5T^{2} \)
83 \( 1 + 605.T + 5.71e5T^{2} \)
89 \( 1 - 218.T + 7.04e5T^{2} \)
97 \( 1 + 782.T + 9.12e5T^{2} \)
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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

−8.327042941640378083140147120702, −7.967059627860772319292417703040, −7.41992028302480036749019169888, −6.37570283519257414353681887382, −5.41003023002926887305493104610, −4.47718289851516277056247245263, −3.72123875561222702666223629492, −2.91869573765371475351539456218, −1.93278482860642723353166887668, −0.46412701501602308708775725105, 0.46412701501602308708775725105, 1.93278482860642723353166887668, 2.91869573765371475351539456218, 3.72123875561222702666223629492, 4.47718289851516277056247245263, 5.41003023002926887305493104610, 6.37570283519257414353681887382, 7.41992028302480036749019169888, 7.967059627860772319292417703040, 8.327042941640378083140147120702

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