Properties

Label 2-2352-1.1-c3-0-37
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 − 0.128·5-s + 9·9-s − 54.0·11-s + 50.2·13-s − 0.385·15-s + 131.·17-s + 91.5·19-s − 179.·23-s − 124.·25-s + 27·27-s − 69.8·29-s + 326.·31-s − 162.·33-s + 301.·37-s + 150.·39-s − 296.·41-s + 144.·43-s − 1.15·45-s − 360.·47-s + 394.·51-s + 1.83·53-s + 6.94·55-s + 274.·57-s − 53.2·59-s + 108.·61-s − 6.45·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.0115·5-s + 0.333·9-s − 1.48·11-s + 1.07·13-s − 0.00664·15-s + 1.87·17-s + 1.10·19-s − 1.62·23-s − 0.999·25-s + 0.192·27-s − 0.447·29-s + 1.89·31-s − 0.854·33-s + 1.34·37-s + 0.618·39-s − 1.12·41-s + 0.511·43-s − 0.00383·45-s − 1.11·47-s + 1.08·51-s + 0.00475·53-s + 0.0170·55-s + 0.637·57-s − 0.117·59-s + 0.226·61-s − 0.0123·65-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

\(L(2)\) \(\approx\) \(2.846644877\)
\(L(\frac12)\) \(\approx\) \(2.846644877\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.128T + 125T^{2} \)
11 \( 1 + 54.0T + 1.33e3T^{2} \)
13 \( 1 - 50.2T + 2.19e3T^{2} \)
17 \( 1 - 131.T + 4.91e3T^{2} \)
19 \( 1 - 91.5T + 6.85e3T^{2} \)
23 \( 1 + 179.T + 1.21e4T^{2} \)
29 \( 1 + 69.8T + 2.43e4T^{2} \)
31 \( 1 - 326.T + 2.97e4T^{2} \)
37 \( 1 - 301.T + 5.06e4T^{2} \)
41 \( 1 + 296.T + 6.89e4T^{2} \)
43 \( 1 - 144.T + 7.95e4T^{2} \)
47 \( 1 + 360.T + 1.03e5T^{2} \)
53 \( 1 - 1.83T + 1.48e5T^{2} \)
59 \( 1 + 53.2T + 2.05e5T^{2} \)
61 \( 1 - 108.T + 2.26e5T^{2} \)
67 \( 1 + 842.T + 3.00e5T^{2} \)
71 \( 1 - 241.T + 3.57e5T^{2} \)
73 \( 1 - 206.T + 3.89e5T^{2} \)
79 \( 1 + 559.T + 4.93e5T^{2} \)
83 \( 1 - 986.T + 5.71e5T^{2} \)
89 \( 1 - 443.T + 7.04e5T^{2} \)
97 \( 1 - 740.T + 9.12e5T^{2} \)
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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

−8.241202196457079442611570594576, −8.047628852300257688429149033770, −7.42326734021609721974967477058, −6.10232230444694308002228141400, −5.62744685557393856436974677125, −4.60678686698642730106025423198, −3.54384156520016549761878239828, −2.95854391324749404882833279229, −1.83508609241373917668012029965, −0.73953391487158070163236334724, 0.73953391487158070163236334724, 1.83508609241373917668012029965, 2.95854391324749404882833279229, 3.54384156520016549761878239828, 4.60678686698642730106025423198, 5.62744685557393856436974677125, 6.10232230444694308002228141400, 7.42326734021609721974967477058, 8.047628852300257688429149033770, 8.241202196457079442611570594576

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