Properties

Label 2-51600-1.1-c1-0-41
Degree $2$
Conductor $51600$
Sign $1$
Analytic cond. $412.028$
Root an. cond. $20.2984$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 2·7-s + 9-s − 11-s − 5·13-s + 6·17-s + 6·19-s + 2·21-s + 9·23-s + 27-s − 6·29-s − 5·31-s − 33-s + 6·37-s − 5·39-s + 3·41-s + 43-s − 13·47-s − 3·49-s + 6·51-s + 13·53-s + 6·57-s + 9·59-s + 8·61-s + 2·63-s − 4·67-s + 9·69-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s + 1.45·17-s + 1.37·19-s + 0.436·21-s + 1.87·23-s + 0.192·27-s − 1.11·29-s − 0.898·31-s − 0.174·33-s + 0.986·37-s − 0.800·39-s + 0.468·41-s + 0.152·43-s − 1.89·47-s − 3/7·49-s + 0.840·51-s + 1.78·53-s + 0.794·57-s + 1.17·59-s + 1.02·61-s + 0.251·63-s − 0.488·67-s + 1.08·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(51600\)    =    \(2^{4} \cdot 3 \cdot 5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(412.028\)
Root analytic conductor: \(20.2984\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{51600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 51600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.799568657\)
\(L(\frac12)\) \(\approx\) \(3.799568657\)
\(L(\frac{3}{2})\) 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 \( 1 \)
43 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 13 T + p T^{2} \)
53 \( 1 - 13 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 17 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−14.58410081487393, −14.29969894361334, −13.35554185745825, −12.98900569594713, −12.59694419958115, −11.83071504979132, −11.43121719265853, −11.02592442467852, −10.11844296315851, −9.814905486212522, −9.379902002302824, −8.745916852234249, −8.097675275624376, −7.542217428587844, −7.342588330326126, −6.772230051766362, −5.581940079163834, −5.338464474885151, −4.887584962195236, −4.082422747386563, −3.324426561776945, −2.886876912877838, −2.172280957408887, −1.385346452704760, −0.6943543051355926, 0.6943543051355926, 1.385346452704760, 2.172280957408887, 2.886876912877838, 3.324426561776945, 4.082422747386563, 4.887584962195236, 5.338464474885151, 5.581940079163834, 6.772230051766362, 7.342588330326126, 7.542217428587844, 8.097675275624376, 8.745916852234249, 9.379902002302824, 9.814905486212522, 10.11844296315851, 11.02592442467852, 11.43121719265853, 11.83071504979132, 12.59694419958115, 12.98900569594713, 13.35554185745825, 14.29969894361334, 14.58410081487393

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