Properties

Label 2-1400-1.1-c1-0-20
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.37·3-s + 7-s + 8.37·9-s + 0.627·11-s + 1.37·13-s − 5.37·17-s + 6.74·19-s + 3.37·21-s − 6.74·23-s + 18.1·27-s + 1.37·29-s − 8·31-s + 2.11·33-s + 2·37-s + 4.62·39-s − 4.74·41-s − 2.74·43-s − 10.1·47-s + 49-s − 18.1·51-s + 0.744·53-s + 22.7·57-s + 8·59-s + 8.74·61-s + 8.37·63-s + 4·67-s − 22.7·69-s + ⋯
L(s)  = 1  + 1.94·3-s + 0.377·7-s + 2.79·9-s + 0.189·11-s + 0.380·13-s − 1.30·17-s + 1.54·19-s + 0.735·21-s − 1.40·23-s + 3.48·27-s + 0.254·29-s − 1.43·31-s + 0.368·33-s + 0.328·37-s + 0.741·39-s − 0.740·41-s − 0.418·43-s − 1.47·47-s + 0.142·49-s − 2.53·51-s + 0.102·53-s + 3.01·57-s + 1.04·59-s + 1.11·61-s + 1.05·63-s + 0.488·67-s − 2.73·69-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: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.592512259\)
\(L(\frac12)\) \(\approx\) \(3.592512259\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.37T + 3T^{2} \)
11 \( 1 - 0.627T + 11T^{2} \)
13 \( 1 - 1.37T + 13T^{2} \)
17 \( 1 + 5.37T + 17T^{2} \)
19 \( 1 - 6.74T + 19T^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 - 1.37T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 4.74T + 41T^{2} \)
43 \( 1 + 2.74T + 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 - 0.744T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 8.74T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 2.11T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 - 3.25T + 89T^{2} \)
97 \( 1 + 18.8T + 97T^{2} \)
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

−9.522499361704848014184694646345, −8.599660450537003361665691671205, −8.189525083927286659334455294686, −7.34381066193211787037707130626, −6.63020756891676730928343959662, −5.18357888397915296882654963675, −4.09603780144951542613436311843, −3.48083297514032757813094558731, −2.38984346767194484336583609876, −1.54427704093114398314754695480, 1.54427704093114398314754695480, 2.38984346767194484336583609876, 3.48083297514032757813094558731, 4.09603780144951542613436311843, 5.18357888397915296882654963675, 6.63020756891676730928343959662, 7.34381066193211787037707130626, 8.189525083927286659334455294686, 8.599660450537003361665691671205, 9.522499361704848014184694646345

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