Properties

Label 2-1400-1.1-c1-0-14
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  + 2.56·3-s + 7-s + 3.56·9-s + 2.12·11-s + 2·13-s − 2.56·17-s − 0.561·19-s + 2.56·21-s + 5.56·23-s + 1.43·27-s + 7.56·29-s − 0.876·31-s + 5.43·33-s − 11.8·37-s + 5.12·39-s − 6.56·41-s + 2.43·43-s + 8.24·47-s + 49-s − 6.56·51-s + 7.12·53-s − 1.43·57-s − 13.3·59-s + 2.87·61-s + 3.56·63-s + 16.1·67-s + 14.2·69-s + ⋯
L(s)  = 1  + 1.47·3-s + 0.377·7-s + 1.18·9-s + 0.640·11-s + 0.554·13-s − 0.621·17-s − 0.128·19-s + 0.558·21-s + 1.15·23-s + 0.276·27-s + 1.40·29-s − 0.157·31-s + 0.946·33-s − 1.94·37-s + 0.820·39-s − 1.02·41-s + 0.371·43-s + 1.20·47-s + 0.142·49-s − 0.918·51-s + 0.978·53-s − 0.190·57-s − 1.74·59-s + 0.368·61-s + 0.448·63-s + 1.96·67-s + 1.71·69-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

\(L(1)\) \(\approx\) \(3.083709222\)
\(L(\frac12)\) \(\approx\) \(3.083709222\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.12T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 2.56T + 17T^{2} \)
19 \( 1 + 0.561T + 19T^{2} \)
23 \( 1 - 5.56T + 23T^{2} \)
29 \( 1 - 7.56T + 29T^{2} \)
31 \( 1 + 0.876T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 + 6.56T + 41T^{2} \)
43 \( 1 - 2.43T + 43T^{2} \)
47 \( 1 - 8.24T + 47T^{2} \)
53 \( 1 - 7.12T + 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 - 2.87T + 61T^{2} \)
67 \( 1 - 16.1T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 6.68T + 79T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−9.218237792899048326957211802772, −8.744118174692972566389473194700, −8.239847596105057719892657208289, −7.20296901163723082148638127686, −6.57094489271139272959239945950, −5.24799940672053052538104844065, −4.21077700994101779347989228616, −3.41357399021207989540056396406, −2.46665012129860963430011425279, −1.37875726556887355406836628188, 1.37875726556887355406836628188, 2.46665012129860963430011425279, 3.41357399021207989540056396406, 4.21077700994101779347989228616, 5.24799940672053052538104844065, 6.57094489271139272959239945950, 7.20296901163723082148638127686, 8.239847596105057719892657208289, 8.744118174692972566389473194700, 9.218237792899048326957211802772

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