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Properties

Label 2-46200-1.1-c1-0-10
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 + 6·13-s + 2·17-s − 2·19-s + 21-s − 27-s − 2·29-s − 4·31-s − 33-s − 8·37-s − 6·39-s + 6·41-s − 8·47-s + 49-s − 2·51-s − 6·53-s + 2·57-s − 10·59-s − 4·61-s − 63-s + 2·67-s + 4·71-s − 10·73-s − 77-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.485·17-s − 0.458·19-s + 0.218·21-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.174·33-s − 1.31·37-s − 0.960·39-s + 0.937·41-s − 1.16·47-s + 1/7·49-s − 0.280·51-s − 0.824·53-s + 0.264·57-s − 1.30·59-s − 0.512·61-s − 0.125·63-s + 0.244·67-s + 0.474·71-s − 1.17·73-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

\(L(1)\) \(\approx\) \(1.552450420\)
\(L(\frac12)\) \(\approx\) \(1.552450420\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.56018313439702, −14.07439624444099, −13.54054486061944, −12.98322245932912, −12.61793794500800, −12.02831843382655, −11.47893161206648, −10.81981084945570, −10.75385536726499, −9.943669664651779, −9.362214372054727, −8.891153136197628, −8.301191557872674, −7.703514548700487, −7.040800829868754, −6.439268145490555, −6.038835583060964, −5.548121296479180, −4.807581097683408, −4.130435922939316, −3.523127401409876, −3.083187498694676, −1.890129554981785, −1.403733894368741, −0.4776025852293053, 0.4776025852293053, 1.403733894368741, 1.890129554981785, 3.083187498694676, 3.523127401409876, 4.130435922939316, 4.807581097683408, 5.548121296479180, 6.038835583060964, 6.439268145490555, 7.040800829868754, 7.703514548700487, 8.301191557872674, 8.891153136197628, 9.362214372054727, 9.943669664651779, 10.75385536726499, 10.81981084945570, 11.47893161206648, 12.02831843382655, 12.61793794500800, 12.98322245932912, 13.54054486061944, 14.07439624444099, 14.56018313439702

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