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Properties

Label 2-46200-1.1-c1-0-15
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 + 13-s + 7·17-s + 3·19-s + 21-s − 27-s − 2·29-s − 4·31-s − 33-s − 3·37-s − 39-s − 9·41-s − 8·47-s + 49-s − 7·51-s + 9·53-s − 3·57-s − 10·59-s + 11·61-s − 63-s − 13·67-s − 71-s − 5·73-s − 77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 1.69·17-s + 0.688·19-s + 0.218·21-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.174·33-s − 0.493·37-s − 0.160·39-s − 1.40·41-s − 1.16·47-s + 1/7·49-s − 0.980·51-s + 1.23·53-s − 0.397·57-s − 1.30·59-s + 1.40·61-s − 0.125·63-s − 1.58·67-s − 0.118·71-s − 0.585·73-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

\(L(1)\) \(\approx\) \(1.714465300\)
\(L(\frac12)\) \(\approx\) \(1.714465300\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−14.70295049978436, −14.13942032017776, −13.37303199025446, −13.22954169889549, −12.39182214250846, −11.94366720710963, −11.74477140211928, −10.95019772045677, −10.45045096676323, −9.910184138724772, −9.560971727970387, −8.837374308761971, −8.305984566134275, −7.487980610748282, −7.268445648886444, −6.488933989683169, −5.956846268095885, −5.414251852998706, −4.977367695130222, −4.130718776865079, −3.372870422201609, −3.178180692702487, −1.946530124043324, −1.330844980877651, −0.5150176116400711, 0.5150176116400711, 1.330844980877651, 1.946530124043324, 3.178180692702487, 3.372870422201609, 4.130718776865079, 4.977367695130222, 5.414251852998706, 5.956846268095885, 6.488933989683169, 7.268445648886444, 7.487980610748282, 8.305984566134275, 8.837374308761971, 9.560971727970387, 9.910184138724772, 10.45045096676323, 10.95019772045677, 11.74477140211928, 11.94366720710963, 12.39182214250846, 13.22954169889549, 13.37303199025446, 14.13942032017776, 14.70295049978436

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