Properties

Label 2-62400-1.1-c1-0-36
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 − 3·7-s + 9-s + 3·11-s + 13-s + 3·17-s − 2·19-s + 3·21-s + 23-s − 27-s + 8·29-s + 4·31-s − 3·33-s − 5·37-s − 39-s − 7·41-s + 2·43-s + 4·47-s + 2·49-s − 3·51-s − 11·53-s + 2·57-s + 4·59-s − 61-s − 3·63-s − 69-s + 5·71-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.727·17-s − 0.458·19-s + 0.654·21-s + 0.208·23-s − 0.192·27-s + 1.48·29-s + 0.718·31-s − 0.522·33-s − 0.821·37-s − 0.160·39-s − 1.09·41-s + 0.304·43-s + 0.583·47-s + 2/7·49-s − 0.420·51-s − 1.51·53-s + 0.264·57-s + 0.520·59-s − 0.128·61-s − 0.377·63-s − 0.120·69-s + 0.593·71-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: \(0\)
Selberg data: \((2,\ 62400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.649549439\)
\(L(\frac12)\) \(\approx\) \(1.649549439\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−14.26965134516852, −13.70869909901364, −13.26327226138480, −12.51653512076678, −12.38876226941536, −11.77666519808668, −11.34723261736424, −10.58233356354525, −10.21639168943354, −9.772026863392263, −9.204936485829937, −8.649045611736202, −8.128807743064077, −7.356930418420853, −6.741825182661449, −6.411058312155213, −6.029862108013176, −5.259078749164333, −4.677307995624336, −4.034427198152939, −3.373384495351351, −2.967393510559684, −1.981085876928486, −1.175497282807092, −0.5046607972339255, 0.5046607972339255, 1.175497282807092, 1.981085876928486, 2.967393510559684, 3.373384495351351, 4.034427198152939, 4.677307995624336, 5.259078749164333, 6.029862108013176, 6.411058312155213, 6.741825182661449, 7.356930418420853, 8.128807743064077, 8.649045611736202, 9.204936485829937, 9.772026863392263, 10.21639168943354, 10.58233356354525, 11.34723261736424, 11.77666519808668, 12.38876226941536, 12.51653512076678, 13.26327226138480, 13.70869909901364, 14.26965134516852

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