Properties

Label 2-62400-1.1-c1-0-117
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 + 4·11-s + 13-s + 6·17-s + 4·21-s − 8·23-s − 27-s − 6·29-s + 4·31-s − 4·33-s − 2·37-s − 39-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s − 6·51-s − 2·53-s + 12·59-s + 2·61-s − 4·63-s + 16·67-s + 8·69-s + 8·71-s + 6·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 1.45·17-s + 0.872·21-s − 1.66·23-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s − 0.328·37-s − 0.160·39-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.840·51-s − 0.274·53-s + 1.56·59-s + 0.256·61-s − 0.503·63-s + 1.95·67-s + 0.963·69-s + 0.949·71-s + 0.702·73-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.39860946572274, −14.05142433357568, −13.46842193124464, −12.91157471567830, −12.45328368174750, −11.96204763264048, −11.70035414607062, −11.00810095333504, −10.30701581468492, −9.798890341364036, −9.671875725038749, −9.058629419425968, −8.184637558139810, −7.900266562564233, −6.936211710002378, −6.590001250567801, −6.279538723095010, −5.518887272907231, −5.224512294532451, −4.084738632336930, −3.712392211930468, −3.382704099655939, −2.398351502065888, −1.592741889705337, −0.8309316507954371, 0, 0.8309316507954371, 1.592741889705337, 2.398351502065888, 3.382704099655939, 3.712392211930468, 4.084738632336930, 5.224512294532451, 5.518887272907231, 6.279538723095010, 6.590001250567801, 6.936211710002378, 7.900266562564233, 8.184637558139810, 9.058629419425968, 9.671875725038749, 9.798890341364036, 10.30701581468492, 11.00810095333504, 11.70035414607062, 11.96204763264048, 12.45328368174750, 12.91157471567830, 13.46842193124464, 14.05142433357568, 14.39860946572274

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