Properties

Label 2-2800-1.1-c1-0-17
Degree $2$
Conductor $2800$
Sign $1$
Analytic cond. $22.3581$
Root an. cond. $4.72843$
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  + 1.23·3-s + 7-s − 1.47·9-s − 4.23·11-s + 3.23·13-s + 6.47·17-s − 4.47·19-s + 1.23·21-s + 1.76·23-s − 5.52·27-s + 5·29-s + 9.70·31-s − 5.23·33-s − 3·37-s + 4.00·39-s + 9.23·41-s + 6.23·43-s − 2·47-s + 49-s + 8.00·51-s + 0.472·53-s − 5.52·57-s + 1.70·59-s + 3.70·61-s − 1.47·63-s + 0.236·67-s + 2.18·69-s + ⋯
L(s)  = 1  + 0.713·3-s + 0.377·7-s − 0.490·9-s − 1.27·11-s + 0.897·13-s + 1.56·17-s − 1.02·19-s + 0.269·21-s + 0.367·23-s − 1.06·27-s + 0.928·29-s + 1.74·31-s − 0.911·33-s − 0.493·37-s + 0.640·39-s + 1.44·41-s + 0.950·43-s − 0.291·47-s + 0.142·49-s + 1.12·51-s + 0.0648·53-s − 0.732·57-s + 0.222·59-s + 0.474·61-s − 0.185·63-s + 0.0288·67-s + 0.262·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(22.3581\)
Root analytic conductor: \(4.72843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.344201102\)
\(L(\frac12)\) \(\approx\) \(2.344201102\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.23T + 3T^{2} \)
11 \( 1 + 4.23T + 11T^{2} \)
13 \( 1 - 3.23T + 13T^{2} \)
17 \( 1 - 6.47T + 17T^{2} \)
19 \( 1 + 4.47T + 19T^{2} \)
23 \( 1 - 1.76T + 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 - 9.70T + 31T^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 - 9.23T + 41T^{2} \)
43 \( 1 - 6.23T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 - 0.472T + 53T^{2} \)
59 \( 1 - 1.70T + 59T^{2} \)
61 \( 1 - 3.70T + 61T^{2} \)
67 \( 1 - 0.236T + 67T^{2} \)
71 \( 1 - 4.70T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 5.70T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + 0.763T + 97T^{2} \)
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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

−8.568704442062272937700776691322, −8.113525760896149142941644389840, −7.64997614619413494519724505227, −6.45374957740923811960333249577, −5.70645458785488601107201515644, −4.93974897452160256386585991469, −3.90976902771277910202252155678, −2.98081772993730869873327434177, −2.34703569094201809767295297023, −0.932723821089314837512743804676, 0.932723821089314837512743804676, 2.34703569094201809767295297023, 2.98081772993730869873327434177, 3.90976902771277910202252155678, 4.93974897452160256386585991469, 5.70645458785488601107201515644, 6.45374957740923811960333249577, 7.64997614619413494519724505227, 8.113525760896149142941644389840, 8.568704442062272937700776691322

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