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Properties

Label 2-46200-1.1-c1-0-35
Degree $2$
Conductor $46200$
Sign $1$
Analytic cond. $368.908$
Root an. cond. $19.2070$
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 + 7-s + 9-s − 11-s + 4·13-s + 4·17-s + 21-s + 6·23-s + 27-s − 6·29-s − 8·31-s − 33-s + 8·37-s + 4·39-s − 2·41-s − 4·43-s − 12·47-s + 49-s + 4·51-s − 2·53-s + 4·59-s + 6·61-s + 63-s − 6·67-s + 6·69-s + 2·73-s − 77-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.970·17-s + 0.218·21-s + 1.25·23-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.174·33-s + 1.31·37-s + 0.640·39-s − 0.312·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.560·51-s − 0.274·53-s + 0.520·59-s + 0.768·61-s + 0.125·63-s − 0.733·67-s + 0.722·69-s + 0.234·73-s − 0.113·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(46200\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(368.908\)
Root analytic conductor: \(19.2070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 46200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.621266082\)
\(L(\frac12)\) \(\approx\) \(3.621266082\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.52320279438362, −14.33473213779271, −13.42415046494346, −13.08337645272115, −12.86400672641416, −12.01598903453749, −11.32542304534153, −11.11899746788857, −10.47404571664382, −9.826961776321033, −9.349702055018788, −8.782169330878379, −8.318746317261675, −7.689595118427389, −7.367559636557487, −6.586222554257261, −5.996685315495093, −5.298930131531585, −4.892037082086887, −3.992265787044150, −3.455954369657134, −3.024547150923238, −2.041463145022577, −1.496081490398746, −0.6751875929101659, 0.6751875929101659, 1.496081490398746, 2.041463145022577, 3.024547150923238, 3.455954369657134, 3.992265787044150, 4.892037082086887, 5.298930131531585, 5.996685315495093, 6.586222554257261, 7.367559636557487, 7.689595118427389, 8.318746317261675, 8.782169330878379, 9.349702055018788, 9.826961776321033, 10.47404571664382, 11.11899746788857, 11.32542304534153, 12.01598903453749, 12.86400672641416, 13.08337645272115, 13.42415046494346, 14.33473213779271, 14.52320279438362

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