Properties

Label 2-1400-1.1-c1-0-21
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 + 13-s + 3·17-s − 4·19-s − 21-s − 2·23-s + 5·27-s − 29-s − 6·31-s + 33-s − 2·37-s − 39-s − 10·41-s − 9·47-s + 49-s − 3·51-s + 14·53-s + 4·57-s + 6·59-s − 4·61-s − 2·63-s − 10·67-s + 2·69-s − 16·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.301·11-s + 0.277·13-s + 0.727·17-s − 0.917·19-s − 0.218·21-s − 0.417·23-s + 0.962·27-s − 0.185·29-s − 1.07·31-s + 0.174·33-s − 0.328·37-s − 0.160·39-s − 1.56·41-s − 1.31·47-s + 1/7·49-s − 0.420·51-s + 1.92·53-s + 0.529·57-s + 0.781·59-s − 0.512·61-s − 0.251·63-s − 1.22·67-s + 0.240·69-s − 1.89·71-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: Trivial
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 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−9.001926936177133387624516807394, −8.425048771595220792826581884601, −7.56241816600185142402203682842, −6.60109526619837542894810840406, −5.73465281057790030728194448161, −5.15295292582642366138384661905, −4.05262910714096236263857673080, −2.95795779308839953337834232357, −1.66123120590211048858003485832, 0, 1.66123120590211048858003485832, 2.95795779308839953337834232357, 4.05262910714096236263857673080, 5.15295292582642366138384661905, 5.73465281057790030728194448161, 6.60109526619837542894810840406, 7.56241816600185142402203682842, 8.425048771595220792826581884601, 9.001926936177133387624516807394

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