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Properties

Label 2-46200-1.1-c1-0-11
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 − 5·13-s + 3·17-s + 7·19-s − 21-s − 4·23-s + 27-s − 4·29-s − 6·31-s − 33-s + 7·37-s − 5·39-s − 3·41-s + 8·43-s − 12·47-s + 49-s + 3·51-s − 7·53-s + 7·57-s + 7·61-s − 63-s + 9·67-s − 4·69-s − 7·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s + 0.727·17-s + 1.60·19-s − 0.218·21-s − 0.834·23-s + 0.192·27-s − 0.742·29-s − 1.07·31-s − 0.174·33-s + 1.15·37-s − 0.800·39-s − 0.468·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.420·51-s − 0.961·53-s + 0.927·57-s + 0.896·61-s − 0.125·63-s + 1.09·67-s − 0.481·69-s − 0.830·71-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

\(L(1)\) \(\approx\) \(2.013915774\)
\(L(\frac12)\) \(\approx\) \(2.013915774\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 7 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−14.55723029961784, −14.25337069625685, −13.62983923588248, −12.97693659516393, −12.71250314618402, −12.04877525705425, −11.61132291099418, −11.03051875856879, −10.16850248154938, −9.889442036083190, −9.411204183623084, −9.050161069413227, −7.954468339875777, −7.841073364837293, −7.307957738645672, −6.714684619019612, −5.901309202334918, −5.351950267189136, −4.869569796478736, −4.046459847831541, −3.418728231130965, −2.899747307476115, −2.235211929499559, −1.495076422238329, −0.4790898910093604, 0.4790898910093604, 1.495076422238329, 2.235211929499559, 2.899747307476115, 3.418728231130965, 4.046459847831541, 4.869569796478736, 5.351950267189136, 5.901309202334918, 6.714684619019612, 7.307957738645672, 7.841073364837293, 7.954468339875777, 9.050161069413227, 9.411204183623084, 9.889442036083190, 10.16850248154938, 11.03051875856879, 11.61132291099418, 12.04877525705425, 12.71250314618402, 12.97693659516393, 13.62983923588248, 14.25337069625685, 14.55723029961784

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