Properties

Label 2-62400-1.1-c1-0-57
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 − 3·7-s + 9-s − 5·11-s + 13-s − 5·17-s + 2·19-s + 3·21-s − 23-s − 27-s − 10·29-s + 2·31-s + 5·33-s − 3·37-s − 39-s − 9·41-s + 4·43-s + 10·47-s + 2·49-s + 5·51-s + 9·53-s − 2·57-s + 11·61-s − 3·63-s + 4·67-s + 69-s − 15·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s + 1/3·9-s − 1.50·11-s + 0.277·13-s − 1.21·17-s + 0.458·19-s + 0.654·21-s − 0.208·23-s − 0.192·27-s − 1.85·29-s + 0.359·31-s + 0.870·33-s − 0.493·37-s − 0.160·39-s − 1.40·41-s + 0.609·43-s + 1.45·47-s + 2/7·49-s + 0.700·51-s + 1.23·53-s − 0.264·57-s + 1.40·61-s − 0.377·63-s + 0.488·67-s + 0.120·69-s − 1.78·71-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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−14.59799914227381, −13.69966923064274, −13.43613296068247, −12.98181837283806, −12.70494574155074, −11.93577557936385, −11.52073583619467, −10.90020835944820, −10.43093931081974, −10.08622190651751, −9.458444042091457, −8.921196412381000, −8.392476018667724, −7.642306445491452, −7.132379241739032, −6.759167103630855, −5.984408768068491, −5.586340299472280, −5.135502211204122, −4.312397822529101, −3.773507717535095, −3.076898953649920, −2.436220206264707, −1.798408033592995, −0.6152955928324311, 0, 0.6152955928324311, 1.798408033592995, 2.436220206264707, 3.076898953649920, 3.773507717535095, 4.312397822529101, 5.135502211204122, 5.586340299472280, 5.984408768068491, 6.759167103630855, 7.132379241739032, 7.642306445491452, 8.392476018667724, 8.921196412381000, 9.458444042091457, 10.08622190651751, 10.43093931081974, 10.90020835944820, 11.52073583619467, 11.93577557936385, 12.70494574155074, 12.98181837283806, 13.43613296068247, 13.69966923064274, 14.59799914227381

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