Properties

Label 2-2352-1.1-c3-0-118
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 15.8·5-s + 9·9-s − 51.8·11-s − 38.8·13-s + 47.5·15-s − 27.3·17-s + 76.5·19-s − 147.·23-s + 125.·25-s + 27·27-s + 240.·29-s − 296.·31-s − 155.·33-s − 161.·37-s − 116.·39-s + 102.·41-s + 328.·43-s + 142.·45-s + 67.9·47-s − 82.0·51-s − 66.4·53-s − 820.·55-s + 229.·57-s − 461.·59-s − 185.·61-s − 615.·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.41·5-s + 0.333·9-s − 1.42·11-s − 0.828·13-s + 0.817·15-s − 0.390·17-s + 0.923·19-s − 1.33·23-s + 1.00·25-s + 0.192·27-s + 1.53·29-s − 1.71·31-s − 0.820·33-s − 0.717·37-s − 0.478·39-s + 0.392·41-s + 1.16·43-s + 0.472·45-s + 0.210·47-s − 0.225·51-s − 0.172·53-s − 2.01·55-s + 0.533·57-s − 1.01·59-s − 0.389·61-s − 1.17·65-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: \(1\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 15.8T + 125T^{2} \)
11 \( 1 + 51.8T + 1.33e3T^{2} \)
13 \( 1 + 38.8T + 2.19e3T^{2} \)
17 \( 1 + 27.3T + 4.91e3T^{2} \)
19 \( 1 - 76.5T + 6.85e3T^{2} \)
23 \( 1 + 147.T + 1.21e4T^{2} \)
29 \( 1 - 240.T + 2.43e4T^{2} \)
31 \( 1 + 296.T + 2.97e4T^{2} \)
37 \( 1 + 161.T + 5.06e4T^{2} \)
41 \( 1 - 102.T + 6.89e4T^{2} \)
43 \( 1 - 328.T + 7.95e4T^{2} \)
47 \( 1 - 67.9T + 1.03e5T^{2} \)
53 \( 1 + 66.4T + 1.48e5T^{2} \)
59 \( 1 + 461.T + 2.05e5T^{2} \)
61 \( 1 + 185.T + 2.26e5T^{2} \)
67 \( 1 + 545.T + 3.00e5T^{2} \)
71 \( 1 - 130.T + 3.57e5T^{2} \)
73 \( 1 + 181.T + 3.89e5T^{2} \)
79 \( 1 - 409.T + 4.93e5T^{2} \)
83 \( 1 - 347.T + 5.71e5T^{2} \)
89 \( 1 + 1.15e3T + 7.04e5T^{2} \)
97 \( 1 + 1.61e3T + 9.12e5T^{2} \)
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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

−8.199057350642650573762978748187, −7.57678933914051344944378316490, −6.74614040972023023945990184685, −5.71181474699281042245901602972, −5.27352194504465938286283802917, −4.27298837167959193524494303464, −2.92919899627255661567124204811, −2.39653278274862668719268326507, −1.53450241016768092346441125728, 0, 1.53450241016768092346441125728, 2.39653278274862668719268326507, 2.92919899627255661567124204811, 4.27298837167959193524494303464, 5.27352194504465938286283802917, 5.71181474699281042245901602972, 6.74614040972023023945990184685, 7.57678933914051344944378316490, 8.199057350642650573762978748187

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