Properties

Label 2-2352-1.1-c3-0-91
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 − 12.4·5-s + 9·9-s + 51.1·11-s − 37.2·13-s − 37.3·15-s − 22.2·17-s + 54.3·19-s − 176.·23-s + 29.9·25-s + 27·27-s + 61.0·29-s + 319.·31-s + 153.·33-s − 315.·37-s − 111.·39-s + 206.·41-s − 339.·43-s − 112.·45-s + 142.·47-s − 66.6·51-s + 310.·53-s − 636.·55-s + 163.·57-s − 281.·59-s + 543.·61-s + 463.·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.11·5-s + 0.333·9-s + 1.40·11-s − 0.794·13-s − 0.642·15-s − 0.316·17-s + 0.656·19-s − 1.60·23-s + 0.239·25-s + 0.192·27-s + 0.391·29-s + 1.85·31-s + 0.809·33-s − 1.39·37-s − 0.458·39-s + 0.784·41-s − 1.20·43-s − 0.371·45-s + 0.440·47-s − 0.182·51-s + 0.803·53-s − 1.55·55-s + 0.378·57-s − 0.621·59-s + 1.14·61-s + 0.884·65-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 \( 1 + 12.4T + 125T^{2} \)
11 \( 1 - 51.1T + 1.33e3T^{2} \)
13 \( 1 + 37.2T + 2.19e3T^{2} \)
17 \( 1 + 22.2T + 4.91e3T^{2} \)
19 \( 1 - 54.3T + 6.85e3T^{2} \)
23 \( 1 + 176.T + 1.21e4T^{2} \)
29 \( 1 - 61.0T + 2.43e4T^{2} \)
31 \( 1 - 319.T + 2.97e4T^{2} \)
37 \( 1 + 315.T + 5.06e4T^{2} \)
41 \( 1 - 206.T + 6.89e4T^{2} \)
43 \( 1 + 339.T + 7.95e4T^{2} \)
47 \( 1 - 142.T + 1.03e5T^{2} \)
53 \( 1 - 310.T + 1.48e5T^{2} \)
59 \( 1 + 281.T + 2.05e5T^{2} \)
61 \( 1 - 543.T + 2.26e5T^{2} \)
67 \( 1 - 479.T + 3.00e5T^{2} \)
71 \( 1 + 1.10e3T + 3.57e5T^{2} \)
73 \( 1 + 239.T + 3.89e5T^{2} \)
79 \( 1 + 1.16e3T + 4.93e5T^{2} \)
83 \( 1 + 2.93T + 5.71e5T^{2} \)
89 \( 1 - 1.27e3T + 7.04e5T^{2} \)
97 \( 1 + 79.0T + 9.12e5T^{2} \)
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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

−8.304531954622511075688480929681, −7.51812530491013254373458673871, −6.90228987848464914718375896584, −6.04849359702424076022020697274, −4.77055690943627350937488380148, −4.09313309230251420716759533840, −3.45963951248356663503328635068, −2.39107797986518227122124598247, −1.22597306770555361957734617123, 0, 1.22597306770555361957734617123, 2.39107797986518227122124598247, 3.45963951248356663503328635068, 4.09313309230251420716759533840, 4.77055690943627350937488380148, 6.04849359702424076022020697274, 6.90228987848464914718375896584, 7.51812530491013254373458673871, 8.304531954622511075688480929681

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