Properties

Label 2-1400-1.1-c1-0-27
Degree $2$
Conductor $1400$
Sign $-1$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·3-s − 7-s − 0.561·9-s + 1.56·11-s − 6.68·13-s − 7.56·17-s − 7.12·19-s − 1.56·21-s − 3.12·23-s − 5.56·27-s + 0.438·29-s + 6.24·31-s + 2.43·33-s + 8.24·37-s − 10.4·39-s − 1.12·41-s + 7.12·43-s − 2.43·47-s + 49-s − 11.8·51-s + 13.1·53-s − 11.1·57-s − 4·59-s − 6.87·61-s + 0.561·63-s − 2.24·67-s − 4.87·69-s + ⋯
L(s)  = 1  + 0.901·3-s − 0.377·7-s − 0.187·9-s + 0.470·11-s − 1.85·13-s − 1.83·17-s − 1.63·19-s − 0.340·21-s − 0.651·23-s − 1.07·27-s + 0.0814·29-s + 1.12·31-s + 0.424·33-s + 1.35·37-s − 1.67·39-s − 0.175·41-s + 1.08·43-s − 0.355·47-s + 0.142·49-s − 1.65·51-s + 1.80·53-s − 1.47·57-s − 0.520·59-s − 0.880·61-s + 0.0707·63-s − 0.274·67-s − 0.587·69-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: no
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 \( 1 - 1.56T + 3T^{2} \)
11 \( 1 - 1.56T + 11T^{2} \)
13 \( 1 + 6.68T + 13T^{2} \)
17 \( 1 + 7.56T + 17T^{2} \)
19 \( 1 + 7.12T + 19T^{2} \)
23 \( 1 + 3.12T + 23T^{2} \)
29 \( 1 - 0.438T + 29T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 - 8.24T + 37T^{2} \)
41 \( 1 + 1.12T + 41T^{2} \)
43 \( 1 - 7.12T + 43T^{2} \)
47 \( 1 + 2.43T + 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 6.87T + 61T^{2} \)
67 \( 1 + 2.24T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 - 0.684T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 5.12T + 89T^{2} \)
97 \( 1 + 1.31T + 97T^{2} \)
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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

−9.098482402185844949142281072268, −8.486227933142193552592465662356, −7.64061143026776784073651096175, −6.74956585590966523874038437379, −6.03279049860851719074815280997, −4.63916836026522710379661577095, −4.06261651262310558125004564964, −2.61990222084121645000190778395, −2.25710122632834324021120631193, 0, 2.25710122632834324021120631193, 2.61990222084121645000190778395, 4.06261651262310558125004564964, 4.63916836026522710379661577095, 6.03279049860851719074815280997, 6.74956585590966523874038437379, 7.64061143026776784073651096175, 8.486227933142193552592465662356, 9.098482402185844949142281072268

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