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Properties

Label 2-46200-1.1-c1-0-24
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 − 8·19-s − 21-s + 4·23-s + 27-s + 2·29-s − 8·31-s + 33-s − 6·37-s + 6·39-s − 2·41-s − 8·43-s + 4·47-s + 49-s + 2·51-s − 2·53-s − 8·57-s − 12·59-s + 10·61-s − 63-s + 12·67-s + 4·69-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 − 1.83·19-s − 0.218·21-s + 0.834·23-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.174·33-s − 0.986·37-s + 0.960·39-s − 0.312·41-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s − 1.05·57-s − 1.56·59-s + 1.28·61-s − 0.125·63-s + 1.46·67-s + 0.481·69-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

\(L(1)\) \(\approx\) \(2.776915377\)
\(L(\frac12)\) \(\approx\) \(2.776915377\)
\(L(\frac{3}{2})\) 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 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−14.56701424290914, −14.19033179387971, −13.50452711605005, −13.02643674445645, −12.82201383319511, −12.11366785976161, −11.43326944056458, −10.92587213142806, −10.44782976481877, −9.994540817057076, −9.165683116009050, −8.725861089639608, −8.531345636918423, −7.817851671735527, −6.994793824232197, −6.712576225896176, −5.996687150732781, −5.546296685926489, −4.598562195491266, −4.097226471881933, −3.368503785633225, −3.131404039057685, −1.977811790925498, −1.585920187930503, −0.5693901618001450, 0.5693901618001450, 1.585920187930503, 1.977811790925498, 3.131404039057685, 3.368503785633225, 4.097226471881933, 4.598562195491266, 5.546296685926489, 5.996687150732781, 6.712576225896176, 6.994793824232197, 7.817851671735527, 8.531345636918423, 8.725861089639608, 9.165683116009050, 9.994540817057076, 10.44782976481877, 10.92587213142806, 11.43326944056458, 12.11366785976161, 12.82201383319511, 13.02643674445645, 13.50452711605005, 14.19033179387971, 14.56701424290914

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