Properties

Label 2-2352-1.1-c3-0-44
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 + 18.9·5-s + 9·9-s − 54.7·11-s + 62.0·13-s − 56.8·15-s + 122.·17-s + 12.5·19-s + 74.4·23-s + 234.·25-s − 27·27-s − 232.·29-s − 10.3·31-s + 164.·33-s − 245.·37-s − 186.·39-s + 238.·41-s + 92.9·43-s + 170.·45-s + 485.·47-s − 367.·51-s − 378.·53-s − 1.03e3·55-s − 37.5·57-s + 182.·59-s + 396.·61-s + 1.17e3·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.69·5-s + 0.333·9-s − 1.50·11-s + 1.32·13-s − 0.978·15-s + 1.74·17-s + 0.151·19-s + 0.674·23-s + 1.87·25-s − 0.192·27-s − 1.48·29-s − 0.0600·31-s + 0.866·33-s − 1.09·37-s − 0.763·39-s + 0.909·41-s + 0.329·43-s + 0.565·45-s + 1.50·47-s − 1.00·51-s − 0.981·53-s − 2.54·55-s − 0.0871·57-s + 0.403·59-s + 0.832·61-s + 2.24·65-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

\(L(2)\) \(\approx\) \(2.878039611\)
\(L(\frac12)\) \(\approx\) \(2.878039611\)
\(L(\frac{5}{2})\) 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 \( 1 - 18.9T + 125T^{2} \)
11 \( 1 + 54.7T + 1.33e3T^{2} \)
13 \( 1 - 62.0T + 2.19e3T^{2} \)
17 \( 1 - 122.T + 4.91e3T^{2} \)
19 \( 1 - 12.5T + 6.85e3T^{2} \)
23 \( 1 - 74.4T + 1.21e4T^{2} \)
29 \( 1 + 232.T + 2.43e4T^{2} \)
31 \( 1 + 10.3T + 2.97e4T^{2} \)
37 \( 1 + 245.T + 5.06e4T^{2} \)
41 \( 1 - 238.T + 6.89e4T^{2} \)
43 \( 1 - 92.9T + 7.95e4T^{2} \)
47 \( 1 - 485.T + 1.03e5T^{2} \)
53 \( 1 + 378.T + 1.48e5T^{2} \)
59 \( 1 - 182.T + 2.05e5T^{2} \)
61 \( 1 - 396.T + 2.26e5T^{2} \)
67 \( 1 - 261.T + 3.00e5T^{2} \)
71 \( 1 - 874.T + 3.57e5T^{2} \)
73 \( 1 - 152.T + 3.89e5T^{2} \)
79 \( 1 + 573.T + 4.93e5T^{2} \)
83 \( 1 + 317.T + 5.71e5T^{2} \)
89 \( 1 + 95.0T + 7.04e5T^{2} \)
97 \( 1 + 1.60e3T + 9.12e5T^{2} \)
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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

−8.739658010500132465782464690833, −7.80504126815167161757594221428, −7.00541759286765564244509733995, −5.95223946637077819540945921463, −5.58642184740769242236728742741, −5.12318318368303619776479128412, −3.67924591047356129973775489867, −2.69293480961935627668399063972, −1.68835392632215501655019969494, −0.817718502839497621070023924337, 0.817718502839497621070023924337, 1.68835392632215501655019969494, 2.69293480961935627668399063972, 3.67924591047356129973775489867, 5.12318318368303619776479128412, 5.58642184740769242236728742741, 5.95223946637077819540945921463, 7.00541759286765564244509733995, 7.80504126815167161757594221428, 8.739658010500132465782464690833

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