Properties

Label 2-1400-1.1-c1-0-23
Degree $2$
Conductor $1400$
Sign $-1$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
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 − 7-s − 2·9-s − 11-s − 6·13-s + 7·17-s + 19-s − 21-s − 8·23-s − 5·27-s − 6·29-s + 4·31-s − 33-s − 8·37-s − 6·39-s − 5·41-s − 6·47-s + 49-s + 7·51-s − 4·53-s + 57-s − 4·59-s + 6·61-s + 2·63-s + 5·67-s − 8·69-s + 14·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 1.66·13-s + 1.69·17-s + 0.229·19-s − 0.218·21-s − 1.66·23-s − 0.962·27-s − 1.11·29-s + 0.718·31-s − 0.174·33-s − 1.31·37-s − 0.960·39-s − 0.780·41-s − 0.875·47-s + 1/7·49-s + 0.980·51-s − 0.549·53-s + 0.132·57-s − 0.520·59-s + 0.768·61-s + 0.251·63-s + 0.610·67-s − 0.963·69-s + 1.66·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 14 T + p T^{2} \)
73 \( 1 + 15 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−9.337230639065845945510700602168, −8.107566967722018383534406259839, −7.82450670059276365611628469847, −6.82404020705380367185989578980, −5.69795078942090655453125020752, −5.09521348715540545761725111624, −3.74075632776133593282338137080, −2.97081882189724539571709885137, −1.98299141225506520923628653836, 0, 1.98299141225506520923628653836, 2.97081882189724539571709885137, 3.74075632776133593282338137080, 5.09521348715540545761725111624, 5.69795078942090655453125020752, 6.82404020705380367185989578980, 7.82450670059276365611628469847, 8.107566967722018383534406259839, 9.337230639065845945510700602168

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