Properties

Label 2-51600-1.1-c1-0-100
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·7-s + 9-s − 11-s + 3·13-s − 17-s − 4·19-s + 2·21-s − 8·23-s − 27-s − 2·29-s − 8·31-s + 33-s − 8·37-s − 3·39-s − 6·41-s + 43-s + 47-s − 3·49-s + 51-s + 6·53-s + 4·57-s − 5·59-s + 4·61-s − 2·63-s − 11·67-s + 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.832·13-s − 0.242·17-s − 0.917·19-s + 0.436·21-s − 1.66·23-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.174·33-s − 1.31·37-s − 0.480·39-s − 0.937·41-s + 0.152·43-s + 0.145·47-s − 3/7·49-s + 0.140·51-s + 0.824·53-s + 0.529·57-s − 0.650·59-s + 0.512·61-s − 0.251·63-s − 1.34·67-s + 0.963·69-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: \(2\)
Selberg data: \((2,\ 51600,\ (\ :1/2),\ 1)\)

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

−14.92438348726782, −14.65044369668991, −13.75579119671266, −13.41834757771046, −12.99870830356928, −12.38568837498927, −12.00113375590024, −11.41720161892009, −10.80499796683924, −10.36128946344396, −10.04457656434574, −9.244371269524564, −8.783554724890091, −8.269713111406903, −7.526784500828080, −7.015771512935515, −6.411969808639891, −5.919314396110145, −5.547720323045132, −4.738832814610570, −4.010230018829797, −3.658508808148833, −2.865486887556766, −1.960894598400226, −1.456533305989053, 0, 0, 1.456533305989053, 1.960894598400226, 2.865486887556766, 3.658508808148833, 4.010230018829797, 4.738832814610570, 5.547720323045132, 5.919314396110145, 6.411969808639891, 7.015771512935515, 7.526784500828080, 8.269713111406903, 8.783554724890091, 9.244371269524564, 10.04457656434574, 10.36128946344396, 10.80499796683924, 11.41720161892009, 12.00113375590024, 12.38568837498927, 12.99870830356928, 13.41834757771046, 13.75579119671266, 14.65044369668991, 14.92438348726782

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