Properties

Label 2-62400-1.1-c1-0-131
Degree $2$
Conductor $62400$
Sign $-1$
Analytic cond. $498.266$
Root an. cond. $22.3218$
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 − 4·7-s + 9-s + 2·11-s − 13-s + 4·17-s + 2·19-s + 4·21-s + 6·23-s − 27-s + 2·29-s + 4·31-s − 2·33-s + 6·37-s + 39-s − 6·41-s + 8·43-s − 8·47-s + 9·49-s − 4·51-s − 10·53-s − 2·57-s − 14·59-s − 10·61-s − 4·63-s + 4·67-s − 6·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.970·17-s + 0.458·19-s + 0.872·21-s + 1.25·23-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.348·33-s + 0.986·37-s + 0.160·39-s − 0.937·41-s + 1.21·43-s − 1.16·47-s + 9/7·49-s − 0.560·51-s − 1.37·53-s − 0.264·57-s − 1.82·59-s − 1.28·61-s − 0.503·63-s + 0.488·67-s − 0.722·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62400\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(498.266\)
Root analytic conductor: \(22.3218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{62400} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 62400,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−14.54713928812735, −13.92969710894589, −13.44637988971697, −12.94820946041598, −12.35811645299477, −12.18962228994877, −11.57237012894134, −10.87205583140829, −10.54410332000820, −9.744225206330357, −9.504326227206326, −9.197040532960773, −8.259505371952697, −7.707868342268392, −7.110995892027460, −6.530152989748058, −6.244778486824041, −5.656765212949288, −4.917909166696540, −4.477161594659441, −3.557805860169832, −3.175237578687956, −2.633503577567804, −1.468492048292472, −0.8785410838147395, 0, 0.8785410838147395, 1.468492048292472, 2.633503577567804, 3.175237578687956, 3.557805860169832, 4.477161594659441, 4.917909166696540, 5.656765212949288, 6.244778486824041, 6.530152989748058, 7.110995892027460, 7.707868342268392, 8.259505371952697, 9.197040532960773, 9.504326227206326, 9.744225206330357, 10.54410332000820, 10.87205583140829, 11.57237012894134, 12.18962228994877, 12.35811645299477, 12.94820946041598, 13.44637988971697, 13.92969710894589, 14.54713928812735

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