Properties

Label 2-62400-1.1-c1-0-105
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 + 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: \(1\)
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 \( 1 + p T^{2} \)
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

−14.53815297796095, −13.71149990272443, −13.56173959763670, −13.04655704121643, −12.40168871746775, −12.01069571869315, −11.69455327763257, −10.76356093275535, −10.56540761696491, −9.926428090180491, −9.390202755376207, −9.136000784602020, −8.323989781984204, −7.697487016509011, −6.982546871697780, −6.779966060748384, −6.097878297241702, −5.516091749809496, −5.227039927001400, −4.142361634787491, −3.852927181963050, −3.040119490537450, −2.624337040002026, −1.557048371891352, −0.7951183347238858, 0, 0.7951183347238858, 1.557048371891352, 2.624337040002026, 3.040119490537450, 3.852927181963050, 4.142361634787491, 5.227039927001400, 5.516091749809496, 6.097878297241702, 6.779966060748384, 6.982546871697780, 7.697487016509011, 8.323989781984204, 9.136000784602020, 9.390202755376207, 9.926428090180491, 10.56540761696491, 10.76356093275535, 11.69455327763257, 12.01069571869315, 12.40168871746775, 13.04655704121643, 13.56173959763670, 13.71149990272443, 14.53815297796095

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