Properties

Label 2-51600-1.1-c1-0-0
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 − 7-s + 9-s + 11-s − 13-s − 4·17-s − 19-s + 21-s − 4·23-s − 27-s − 9·29-s − 2·31-s − 33-s − 2·37-s + 39-s − 8·41-s − 43-s − 11·47-s − 6·49-s + 4·51-s − 4·53-s + 57-s + 12·59-s − 63-s + 2·67-s + 4·69-s − 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 0.970·17-s − 0.229·19-s + 0.218·21-s − 0.834·23-s − 0.192·27-s − 1.67·29-s − 0.359·31-s − 0.174·33-s − 0.328·37-s + 0.160·39-s − 1.24·41-s − 0.152·43-s − 1.60·47-s − 6/7·49-s + 0.560·51-s − 0.549·53-s + 0.132·57-s + 1.56·59-s − 0.125·63-s + 0.244·67-s + 0.481·69-s − 1.42·71-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

\(L(1)\) \(\approx\) \(0.07365720554\)
\(L(\frac12)\) \(\approx\) \(0.07365720554\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−14.61636125133659, −13.93657876762619, −13.27867076625848, −12.98910312158544, −12.50779783492741, −11.82016872588768, −11.35051612681574, −11.11349677357958, −10.21304955730671, −9.953652228628598, −9.375504202329800, −8.748595528584038, −8.254023035737792, −7.531539538410844, −6.932048999848235, −6.570306406289828, −5.930983077985868, −5.390430190774213, −4.774638436576435, −4.114699093505135, −3.609079304468307, −2.839257247437655, −1.928956412416844, −1.499115711057900, −0.09443830582451543, 0.09443830582451543, 1.499115711057900, 1.928956412416844, 2.839257247437655, 3.609079304468307, 4.114699093505135, 4.774638436576435, 5.390430190774213, 5.930983077985868, 6.570306406289828, 6.932048999848235, 7.531539538410844, 8.254023035737792, 8.748595528584038, 9.375504202329800, 9.953652228628598, 10.21304955730671, 11.11349677357958, 11.35051612681574, 11.82016872588768, 12.50779783492741, 12.98910312158544, 13.27867076625848, 13.93657876762619, 14.61636125133659

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