Properties

Label 2-2352-1.1-c1-0-19
Degree $2$
Conductor $2352$
Sign $1$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4.27·5-s + 9-s + 4.27·11-s + 1.27·13-s − 4.27·15-s + 4·17-s + 1.27·19-s − 4·23-s + 13.2·25-s − 27-s − 2.27·29-s − 31-s − 4.27·33-s + 5.27·37-s − 1.27·39-s − 10.5·41-s + 7.27·43-s + 4.27·45-s − 6·47-s − 4·51-s + 1.72·53-s + 18.2·55-s − 1.27·57-s + 6.27·59-s − 10·61-s + 5.45·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.91·5-s + 0.333·9-s + 1.28·11-s + 0.353·13-s − 1.10·15-s + 0.970·17-s + 0.292·19-s − 0.834·23-s + 2.65·25-s − 0.192·27-s − 0.422·29-s − 0.179·31-s − 0.744·33-s + 0.867·37-s − 0.204·39-s − 1.64·41-s + 1.10·43-s + 0.637·45-s − 0.875·47-s − 0.560·51-s + 0.236·53-s + 2.46·55-s − 0.168·57-s + 0.816·59-s − 1.28·61-s + 0.676·65-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.488345946\)
\(L(\frac12)\) \(\approx\) \(2.488345946\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 \)
good5 \( 1 - 4.27T + 5T^{2} \)
11 \( 1 - 4.27T + 11T^{2} \)
13 \( 1 - 1.27T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 1.27T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 2.27T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 - 5.27T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 7.27T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 - 1.72T + 53T^{2} \)
59 \( 1 - 6.27T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 7.27T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 3.27T + 73T^{2} \)
79 \( 1 - 3.54T + 79T^{2} \)
83 \( 1 + 0.274T + 83T^{2} \)
89 \( 1 + 4.54T + 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−9.282567009745935289781846600577, −8.360765315786156558074331608821, −7.18666918250348563056368236375, −6.41479555205882377247980321841, −5.89200406639046909124189164531, −5.32697921919298232207247039336, −4.26360078687763451284615582494, −3.13465765970848089253992018844, −1.86511148059292362865388348851, −1.19070059661617080856778409750, 1.19070059661617080856778409750, 1.86511148059292362865388348851, 3.13465765970848089253992018844, 4.26360078687763451284615582494, 5.32697921919298232207247039336, 5.89200406639046909124189164531, 6.41479555205882377247980321841, 7.18666918250348563056368236375, 8.360765315786156558074331608821, 9.282567009745935289781846600577

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