Properties

Label 2-51600-1.1-c1-0-43
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s − 4·11-s − 6·13-s + 2·17-s + 4·23-s − 27-s + 2·29-s − 8·31-s + 4·33-s − 2·37-s + 6·39-s + 10·41-s + 43-s − 4·47-s − 7·49-s − 2·51-s + 14·53-s + 4·59-s − 2·61-s − 4·67-s − 4·69-s − 14·73-s − 8·79-s + 81-s + 8·83-s − 2·87-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.485·17-s + 0.834·23-s − 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s − 0.328·37-s + 0.960·39-s + 1.56·41-s + 0.152·43-s − 0.583·47-s − 49-s − 0.280·51-s + 1.92·53-s + 0.520·59-s − 0.256·61-s − 0.488·67-s − 0.481·69-s − 1.63·73-s − 0.900·79-s + 1/9·81-s + 0.878·83-s − 0.214·87-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: \(1\)
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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−14.86920704938842, −14.32107406239994, −13.68579918628535, −12.98184271457956, −12.66388906438878, −12.37346016781698, −11.47711612181368, −11.34042646453365, −10.46386500672968, −10.21172588722948, −9.723500062446415, −9.040750196844141, −8.514042920758942, −7.612432583678788, −7.408285245316778, −7.002820332883651, −6.021282616000825, −5.640224068173809, −4.923241252232568, −4.771619521612256, −3.850283960524540, −3.027390508937102, −2.510339875303944, −1.798805762203548, −0.7532976505452261, 0, 0.7532976505452261, 1.798805762203548, 2.510339875303944, 3.027390508937102, 3.850283960524540, 4.771619521612256, 4.923241252232568, 5.640224068173809, 6.021282616000825, 7.002820332883651, 7.408285245316778, 7.612432583678788, 8.514042920758942, 9.040750196844141, 9.723500062446415, 10.21172588722948, 10.46386500672968, 11.34042646453365, 11.47711612181368, 12.37346016781698, 12.66388906438878, 12.98184271457956, 13.68579918628535, 14.32107406239994, 14.86920704938842

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