Properties

Label 2-62400-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s − 13-s − 5·19-s + 4·21-s − 27-s − 3·29-s − 4·31-s + 7·37-s + 39-s + 3·41-s − 2·43-s + 9·47-s + 9·49-s − 9·53-s + 5·57-s − 6·59-s − 8·61-s − 4·63-s − 5·67-s − 3·71-s − 4·73-s + 11·79-s + 81-s + 6·83-s + 3·87-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 1.14·19-s + 0.872·21-s − 0.192·27-s − 0.557·29-s − 0.718·31-s + 1.15·37-s + 0.160·39-s + 0.468·41-s − 0.304·43-s + 1.31·47-s + 9/7·49-s − 1.23·53-s + 0.662·57-s − 0.781·59-s − 1.02·61-s − 0.503·63-s − 0.610·67-s − 0.356·71-s − 0.468·73-s + 1.23·79-s + 1/9·81-s + 0.658·83-s + 0.321·87-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: \(0\)
Selberg data: \((2,\ 62400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2973744975\)
\(L(\frac12)\) \(\approx\) \(0.2973744975\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.30866547002786, −13.47564212710571, −13.28708371555753, −12.65340171690906, −12.35774061457731, −11.90983338085365, −10.99901612472245, −10.83116355696825, −10.23472510975716, −9.613903610652446, −9.258969175534280, −8.821015817022692, −7.841257566540626, −7.550076645720394, −6.721321916614084, −6.433040900040513, −5.916331494817457, −5.402859842342877, −4.560994133938580, −4.096131074315853, −3.439136463410053, −2.781437205580997, −2.141541679393143, −1.214121109407429, −0.2001157127338497, 0.2001157127338497, 1.214121109407429, 2.141541679393143, 2.781437205580997, 3.439136463410053, 4.096131074315853, 4.560994133938580, 5.402859842342877, 5.916331494817457, 6.433040900040513, 6.721321916614084, 7.550076645720394, 7.841257566540626, 8.821015817022692, 9.258969175534280, 9.613903610652446, 10.23472510975716, 10.83116355696825, 10.99901612472245, 11.90983338085365, 12.35774061457731, 12.65340171690906, 13.28708371555753, 13.47564212710571, 14.30866547002786

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