Properties

Label 2-5700-1.1-c1-0-51
Degree $2$
Conductor $5700$
Sign $-1$
Analytic cond. $45.5147$
Root an. cond. $6.74646$
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  + 3-s + 2.60·7-s + 9-s − 4.60·11-s − 4.60·13-s − 2·17-s − 19-s + 2.60·21-s + 2·23-s + 27-s + 2.60·29-s + 4·31-s − 4.60·33-s − 3.39·37-s − 4.60·39-s + 6.60·41-s − 10.6·43-s − 6·47-s − 0.211·49-s − 2·51-s − 57-s + 5.21·59-s − 7.21·61-s + 2.60·63-s − 4·67-s + 2·69-s − 9.21·71-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.984·7-s + 0.333·9-s − 1.38·11-s − 1.27·13-s − 0.485·17-s − 0.229·19-s + 0.568·21-s + 0.417·23-s + 0.192·27-s + 0.483·29-s + 0.718·31-s − 0.801·33-s − 0.558·37-s − 0.737·39-s + 1.03·41-s − 1.61·43-s − 0.875·47-s − 0.0301·49-s − 0.280·51-s − 0.132·57-s + 0.678·59-s − 0.923·61-s + 0.328·63-s − 0.488·67-s + 0.240·69-s − 1.09·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5700\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(45.5147\)
Root analytic conductor: \(6.74646\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5700,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 - T \)
5 \( 1 \)
19 \( 1 + T \)
good7 \( 1 - 2.60T + 7T^{2} \)
11 \( 1 + 4.60T + 11T^{2} \)
13 \( 1 + 4.60T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 - 2.60T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 3.39T + 37T^{2} \)
41 \( 1 - 6.60T + 41T^{2} \)
43 \( 1 + 10.6T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 5.21T + 59T^{2} \)
61 \( 1 + 7.21T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 9.21T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 - 6.60T + 89T^{2} \)
97 \( 1 + 16.6T + 97T^{2} \)
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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

−8.005496798035787714169369289047, −7.21631094202140810468760249785, −6.50850667230466942289921119201, −5.29336658304710231161040098333, −4.92209435464879378526023176096, −4.22655493173202345481604399752, −2.97138795116333100930575535638, −2.46177021673501135857725024930, −1.53899881658505912435319864963, 0, 1.53899881658505912435319864963, 2.46177021673501135857725024930, 2.97138795116333100930575535638, 4.22655493173202345481604399752, 4.92209435464879378526023176096, 5.29336658304710231161040098333, 6.50850667230466942289921119201, 7.21631094202140810468760249785, 8.005496798035787714169369289047

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