Properties

Label 2-51600-1.1-c1-0-33
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 − 2·13-s + 6·17-s − 8·19-s + 2·21-s + 6·23-s + 27-s − 6·29-s + 4·31-s − 8·37-s − 2·39-s + 6·41-s + 43-s + 6·47-s − 3·49-s + 6·51-s + 12·53-s − 8·57-s − 4·61-s + 2·63-s − 4·67-s + 6·69-s + 10·73-s − 8·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.554·13-s + 1.45·17-s − 1.83·19-s + 0.436·21-s + 1.25·23-s + 0.192·27-s − 1.11·29-s + 0.718·31-s − 1.31·37-s − 0.320·39-s + 0.937·41-s + 0.152·43-s + 0.875·47-s − 3/7·49-s + 0.840·51-s + 1.64·53-s − 1.05·57-s − 0.512·61-s + 0.251·63-s − 0.488·67-s + 0.722·69-s + 1.17·73-s − 0.900·79-s + 1/9·81-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

\(L(1)\) \(\approx\) \(3.338260171\)
\(L(\frac12)\) \(\approx\) \(3.338260171\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.65840670391528, −14.02735987164027, −13.57506697609410, −12.85557911849684, −12.54797119162979, −12.01087801746397, −11.39896900296429, −10.75418471250996, −10.43313793883819, −9.840122303693080, −9.119599814335293, −8.813362069249086, −8.123458945787847, −7.772870533126824, −7.125568987469382, −6.701570739124096, −5.766986863363831, −5.377899746275897, −4.636102675338868, −4.159231556399081, −3.463629999229338, −2.769999825900936, −2.129723903573875, −1.490630610598595, −0.6217978113793413, 0.6217978113793413, 1.490630610598595, 2.129723903573875, 2.769999825900936, 3.463629999229338, 4.159231556399081, 4.636102675338868, 5.377899746275897, 5.766986863363831, 6.701570739124096, 7.125568987469382, 7.772870533126824, 8.123458945787847, 8.813362069249086, 9.119599814335293, 9.840122303693080, 10.43313793883819, 10.75418471250996, 11.39896900296429, 12.01087801746397, 12.54797119162979, 12.85557911849684, 13.57506697609410, 14.02735987164027, 14.65840670391528

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