Properties

Label 2-1400-1.1-c1-0-16
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.12·3-s − 7-s + 6.76·9-s + 2.48·11-s − 4.15·13-s + 5.76·17-s − 1.60·19-s − 3.12·21-s + 7.28·23-s + 11.7·27-s + 1.45·29-s − 2.24·31-s + 7.76·33-s − 6·37-s − 12.9·39-s + 11.2·41-s − 5.28·43-s + 3.45·47-s + 49-s + 18.0·51-s − 9.21·53-s − 5.03·57-s − 5.92·59-s + 5.35·61-s − 6.76·63-s − 7.52·67-s + 22.7·69-s + ⋯
L(s)  = 1  + 1.80·3-s − 0.377·7-s + 2.25·9-s + 0.749·11-s − 1.15·13-s + 1.39·17-s − 0.369·19-s − 0.681·21-s + 1.51·23-s + 2.26·27-s + 0.270·29-s − 0.404·31-s + 1.35·33-s − 0.986·37-s − 2.07·39-s + 1.76·41-s − 0.805·43-s + 0.503·47-s + 0.142·49-s + 2.52·51-s − 1.26·53-s − 0.666·57-s − 0.770·59-s + 0.686·61-s − 0.852·63-s − 0.919·67-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

\(L(1)\) \(\approx\) \(3.239012867\)
\(L(\frac12)\) \(\approx\) \(3.239012867\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.12T + 3T^{2} \)
11 \( 1 - 2.48T + 11T^{2} \)
13 \( 1 + 4.15T + 13T^{2} \)
17 \( 1 - 5.76T + 17T^{2} \)
19 \( 1 + 1.60T + 19T^{2} \)
23 \( 1 - 7.28T + 23T^{2} \)
29 \( 1 - 1.45T + 29T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 + 5.28T + 43T^{2} \)
47 \( 1 - 3.45T + 47T^{2} \)
53 \( 1 + 9.21T + 53T^{2} \)
59 \( 1 + 5.92T + 59T^{2} \)
61 \( 1 - 5.35T + 61T^{2} \)
67 \( 1 + 7.52T + 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 - 7.28T + 73T^{2} \)
79 \( 1 - 16.9T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + 2.73T + 97T^{2} \)
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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

−9.409344069834454129631226393445, −8.901213072744211247888683387755, −7.956467123206446196136747544306, −7.35235238807244672391641168005, −6.62310445893279308196615686653, −5.22010502321518347905447888820, −4.16965028885278454720126471453, −3.28623503642047064414362320420, −2.62002873284730933572755918240, −1.38865274016735659785868412142, 1.38865274016735659785868412142, 2.62002873284730933572755918240, 3.28623503642047064414362320420, 4.16965028885278454720126471453, 5.22010502321518347905447888820, 6.62310445893279308196615686653, 7.35235238807244672391641168005, 7.956467123206446196136747544306, 8.901213072744211247888683387755, 9.409344069834454129631226393445

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