Properties

Label 2-2352-1.1-c1-0-6
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·5-s + 9-s − 3·11-s + 4·13-s − 3·15-s − 4·19-s + 4·25-s + 27-s + 9·29-s − 31-s − 3·33-s + 8·37-s + 4·39-s + 10·43-s − 3·45-s − 6·47-s − 3·53-s + 9·55-s − 4·57-s + 3·59-s + 10·61-s − 12·65-s + 10·67-s + 6·71-s − 2·73-s + 4·75-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.904·11-s + 1.10·13-s − 0.774·15-s − 0.917·19-s + 4/5·25-s + 0.192·27-s + 1.67·29-s − 0.179·31-s − 0.522·33-s + 1.31·37-s + 0.640·39-s + 1.52·43-s − 0.447·45-s − 0.875·47-s − 0.412·53-s + 1.21·55-s − 0.529·57-s + 0.390·59-s + 1.28·61-s − 1.48·65-s + 1.22·67-s + 0.712·71-s − 0.234·73-s + 0.461·75-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.571673503\)
\(L(\frac12)\) \(\approx\) \(1.571673503\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−8.612494388734689694991205728338, −8.280663412385852718756122161018, −7.67771382302582731127739747771, −6.81063596635974822105393115253, −5.94625450706529550023552218921, −4.73257716458516408324194799472, −4.07941252898153916837817098225, −3.28449031719662890486285908708, −2.35337198591984192566335450633, −0.78012318991079075998632650767, 0.78012318991079075998632650767, 2.35337198591984192566335450633, 3.28449031719662890486285908708, 4.07941252898153916837817098225, 4.73257716458516408324194799472, 5.94625450706529550023552218921, 6.81063596635974822105393115253, 7.67771382302582731127739747771, 8.280663412385852718756122161018, 8.612494388734689694991205728338

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