Properties

Label 2-2352-1.1-c1-0-32
Degree $2$
Conductor $2352$
Sign $-1$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s + 9-s − 2·11-s − 3·13-s − 2·15-s + 8·17-s + 19-s − 8·23-s − 25-s + 27-s + 4·29-s − 3·31-s − 2·33-s − 37-s − 3·39-s + 6·41-s − 11·43-s − 2·45-s − 6·47-s + 8·51-s − 12·53-s + 4·55-s + 57-s − 4·59-s − 6·61-s + 6·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.603·11-s − 0.832·13-s − 0.516·15-s + 1.94·17-s + 0.229·19-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.538·31-s − 0.348·33-s − 0.164·37-s − 0.480·39-s + 0.937·41-s − 1.67·43-s − 0.298·45-s − 0.875·47-s + 1.12·51-s − 1.64·53-s + 0.539·55-s + 0.132·57-s − 0.520·59-s − 0.768·61-s + 0.744·65-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: yes
Arithmetic: yes
Character: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−8.304389845549705419721349874798, −7.80305217239486941411038316758, −7.48792818245316224281758841542, −6.29726913873598404702318801092, −5.34353355016647543832466490803, −4.49676339846623379295899616014, −3.55326335635197764003572865473, −2.90196711465655366821264757704, −1.63793208380212130696209039180, 0, 1.63793208380212130696209039180, 2.90196711465655366821264757704, 3.55326335635197764003572865473, 4.49676339846623379295899616014, 5.34353355016647543832466490803, 6.29726913873598404702318801092, 7.48792818245316224281758841542, 7.80305217239486941411038316758, 8.304389845549705419721349874798

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