Properties

Label 2-1400-1.1-c1-0-22
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  + 7-s − 3·9-s + 11-s − 2·13-s − 4·17-s − 2·19-s − 5·23-s + 29-s − 2·31-s − 3·37-s + 12·41-s − 11·43-s − 2·47-s + 49-s − 6·53-s − 10·59-s + 4·61-s − 3·63-s − 67-s − 3·71-s + 77-s − 9·79-s + 9·81-s + 2·83-s − 6·89-s − 2·91-s − 14·97-s + ⋯
L(s)  = 1  + 0.377·7-s − 9-s + 0.301·11-s − 0.554·13-s − 0.970·17-s − 0.458·19-s − 1.04·23-s + 0.185·29-s − 0.359·31-s − 0.493·37-s + 1.87·41-s − 1.67·43-s − 0.291·47-s + 1/7·49-s − 0.824·53-s − 1.30·59-s + 0.512·61-s − 0.377·63-s − 0.122·67-s − 0.356·71-s + 0.113·77-s − 1.01·79-s + 81-s + 0.219·83-s − 0.635·89-s − 0.209·91-s − 1.42·97-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 \( 1 + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−9.063732721407264098804255752994, −8.410319456039944378168425749912, −7.64853346454933513923745764234, −6.62385461300390207913727757851, −5.89185191436335680897716961439, −4.92280621580036592880794499259, −4.06299815122178043432589638506, −2.85830357112156475553686029481, −1.85727000132990006956153760338, 0, 1.85727000132990006956153760338, 2.85830357112156475553686029481, 4.06299815122178043432589638506, 4.92280621580036592880794499259, 5.89185191436335680897716961439, 6.62385461300390207913727757851, 7.64853346454933513923745764234, 8.410319456039944378168425749912, 9.063732721407264098804255752994

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