Properties

Label 2-1400-1.1-c1-0-17
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  − 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: \(1\)
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 \( 1 - 2.12T + 11T^{2} \)
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

−9.448236076997645326736019268313, −8.268059032052610293781751519605, −7.36629948391706331494942942697, −6.33825622347197078298359905109, −6.09437465928392248430617542934, −4.97851328967325605875195873002, −4.29528314099125389739771012556, −3.00158667861434916034148995806, −1.39586011083815699631381559088, 0, 1.39586011083815699631381559088, 3.00158667861434916034148995806, 4.29528314099125389739771012556, 4.97851328967325605875195873002, 6.09437465928392248430617542934, 6.33825622347197078298359905109, 7.36629948391706331494942942697, 8.268059032052610293781751519605, 9.448236076997645326736019268313

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