Properties

Label 2-1400-1.1-c1-0-15
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.56·3-s − 7-s + 3.56·9-s − 2.56·11-s + 5.68·13-s − 3.43·17-s + 1.12·19-s + 2.56·21-s + 5.12·23-s − 1.43·27-s + 4.56·29-s − 10.2·31-s + 6.56·33-s − 8.24·37-s − 14.5·39-s + 7.12·41-s − 1.12·43-s − 6.56·47-s + 49-s + 8.80·51-s + 4.87·53-s − 2.87·57-s − 4·59-s − 15.1·61-s − 3.56·63-s + 14.2·67-s − 13.1·69-s + ⋯
L(s)  = 1  − 1.47·3-s − 0.377·7-s + 1.18·9-s − 0.772·11-s + 1.57·13-s − 0.833·17-s + 0.257·19-s + 0.558·21-s + 1.06·23-s − 0.276·27-s + 0.847·29-s − 1.84·31-s + 1.14·33-s − 1.35·37-s − 2.33·39-s + 1.11·41-s − 0.171·43-s − 0.957·47-s + 0.142·49-s + 1.23·51-s + 0.669·53-s − 0.381·57-s − 0.520·59-s − 1.93·61-s − 0.448·63-s + 1.74·67-s − 1.57·69-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 \( 1 + 2.56T + 3T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
13 \( 1 - 5.68T + 13T^{2} \)
17 \( 1 + 3.43T + 17T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 - 5.12T + 23T^{2} \)
29 \( 1 - 4.56T + 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 - 7.12T + 41T^{2} \)
43 \( 1 + 1.12T + 43T^{2} \)
47 \( 1 + 6.56T + 47T^{2} \)
53 \( 1 - 4.87T + 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 15.1T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 3.12T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.167696575589424661994937861020, −8.449478658662268270449685917465, −7.23744510400946632512343354789, −6.58363209623201446301267933068, −5.79246375377138578107140330272, −5.18923717007332597449709250332, −4.18158383396793899454387122321, −3.03602382277812977809384611934, −1.38624561847245772024537446681, 0, 1.38624561847245772024537446681, 3.03602382277812977809384611934, 4.18158383396793899454387122321, 5.18923717007332597449709250332, 5.79246375377138578107140330272, 6.58363209623201446301267933068, 7.23744510400946632512343354789, 8.449478658662268270449685917465, 9.167696575589424661994937861020

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