Properties

Label 2-1456-1.1-c1-0-18
Degree $2$
Conductor $1456$
Sign $1$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
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  + 2·3-s + 5-s + 7-s + 9-s + 4·11-s + 13-s + 2·15-s − 2·17-s + 19-s + 2·21-s + 7·23-s − 4·25-s − 4·27-s − 5·29-s + 9·31-s + 8·33-s + 35-s − 2·37-s + 2·39-s + 2·41-s − 43-s + 45-s − 9·47-s + 49-s − 4·51-s + 3·53-s + 4·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.516·15-s − 0.485·17-s + 0.229·19-s + 0.436·21-s + 1.45·23-s − 4/5·25-s − 0.769·27-s − 0.928·29-s + 1.61·31-s + 1.39·33-s + 0.169·35-s − 0.328·37-s + 0.320·39-s + 0.312·41-s − 0.152·43-s + 0.149·45-s − 1.31·47-s + 1/7·49-s − 0.560·51-s + 0.412·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.989953146\)
\(L(\frac12)\) \(\approx\) \(2.989953146\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
13 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
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

−9.384639507251552777367553326751, −8.735471227383272204170095485479, −8.144388668476396997222119983247, −7.14918000366524593178728481657, −6.37684020436893069227778189405, −5.34298608831346054320252005808, −4.23622923362218866686267765866, −3.39334742454479681717934942712, −2.38361565588352713517870291579, −1.35836937709627763557977801195, 1.35836937709627763557977801195, 2.38361565588352713517870291579, 3.39334742454479681717934942712, 4.23622923362218866686267765866, 5.34298608831346054320252005808, 6.37684020436893069227778189405, 7.14918000366524593178728481657, 8.144388668476396997222119983247, 8.735471227383272204170095485479, 9.384639507251552777367553326751

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