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Properties

Label 2-46200-1.1-c1-0-17
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 − 2·13-s + 6·17-s − 4·19-s − 21-s − 4·23-s + 27-s + 2·29-s − 4·31-s + 33-s + 2·37-s − 2·39-s + 2·41-s − 8·43-s + 49-s + 6·51-s + 6·53-s − 4·57-s − 4·59-s + 2·61-s − 63-s − 12·67-s − 4·69-s + 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.834·23-s + 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.174·33-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 1.21·43-s + 1/7·49-s + 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.125·63-s − 1.46·67-s − 0.481·69-s + 1.42·71-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

\(L(1)\) \(\approx\) \(2.418140030\)
\(L(\frac12)\) \(\approx\) \(2.418140030\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−14.54098408898400, −14.20909644085438, −13.64046029385897, −13.07560258195270, −12.55823664894331, −12.07645797455739, −11.73421221695019, −10.80208905987694, −10.40522868858233, −9.837586983900114, −9.431686721645321, −8.879107241737834, −8.169960810663045, −7.828974402094279, −7.235574780687806, −6.544408362684901, −6.117024191210039, −5.334405534051383, −4.811424140604774, −3.918521582914067, −3.628983019354303, −2.837736142187100, −2.204797573153098, −1.491610857051830, −0.5315013843269370, 0.5315013843269370, 1.491610857051830, 2.204797573153098, 2.837736142187100, 3.628983019354303, 3.918521582914067, 4.811424140604774, 5.334405534051383, 6.117024191210039, 6.544408362684901, 7.235574780687806, 7.828974402094279, 8.169960810663045, 8.879107241737834, 9.431686721645321, 9.837586983900114, 10.40522868858233, 10.80208905987694, 11.73421221695019, 12.07645797455739, 12.55823664894331, 13.07560258195270, 13.64046029385897, 14.20909644085438, 14.54098408898400

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