Properties

Label 2-62400-1.1-c1-0-104
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 + 11-s − 13-s − 5·17-s + 8·19-s + 3·21-s − 27-s − 29-s + 3·31-s − 33-s + 8·37-s + 39-s − 2·41-s − 8·43-s − 11·47-s + 2·49-s + 5·51-s + 11·53-s − 8·57-s − 5·59-s − 61-s − 3·63-s − 3·67-s + 16·71-s − 4·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s − 1.21·17-s + 1.83·19-s + 0.654·21-s − 0.192·27-s − 0.185·29-s + 0.538·31-s − 0.174·33-s + 1.31·37-s + 0.160·39-s − 0.312·41-s − 1.21·43-s − 1.60·47-s + 2/7·49-s + 0.700·51-s + 1.51·53-s − 1.05·57-s − 0.650·59-s − 0.128·61-s − 0.377·63-s − 0.366·67-s + 1.89·71-s − 0.468·73-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: Trivial
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 \( 1 - T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−14.56677479304111, −13.81517179214112, −13.38562078737937, −13.16318243907473, −12.39612647955220, −12.03123366657443, −11.44562669406362, −11.15760666094344, −10.35564422447067, −9.904207275771850, −9.423288006471314, −9.164218164748209, −8.238603561217949, −7.781582525058372, −7.011264871286580, −6.618691867548970, −6.298227233741467, −5.471819030469926, −5.071631672620880, −4.372469473024091, −3.715796166030284, −3.114959670814462, −2.524286064657432, −1.607785814138116, −0.7902111168218075, 0, 0.7902111168218075, 1.607785814138116, 2.524286064657432, 3.114959670814462, 3.715796166030284, 4.372469473024091, 5.071631672620880, 5.471819030469926, 6.298227233741467, 6.618691867548970, 7.011264871286580, 7.781582525058372, 8.238603561217949, 9.164218164748209, 9.423288006471314, 9.904207275771850, 10.35564422447067, 11.15760666094344, 11.44562669406362, 12.03123366657443, 12.39612647955220, 13.16318243907473, 13.38562078737937, 13.81517179214112, 14.56677479304111

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