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Properties

Label 2-46200-1.1-c1-0-49
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s − 11-s − 6·13-s + 5·17-s − 5·19-s − 21-s − 23-s − 27-s + 3·29-s + 33-s + 6·37-s + 6·39-s + 4·41-s − 11·43-s + 4·47-s + 49-s − 5·51-s − 53-s + 5·57-s + 15·59-s − 15·61-s + 63-s + 10·67-s + 69-s + 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 1.21·17-s − 1.14·19-s − 0.218·21-s − 0.208·23-s − 0.192·27-s + 0.557·29-s + 0.174·33-s + 0.986·37-s + 0.960·39-s + 0.624·41-s − 1.67·43-s + 0.583·47-s + 1/7·49-s − 0.700·51-s − 0.137·53-s + 0.662·57-s + 1.95·59-s − 1.92·61-s + 0.125·63-s + 1.22·67-s + 0.120·69-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: \(1\)
Selberg data: \((2,\ 46200,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + 15 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−14.83480658090699, −14.46357932849959, −13.94192682504980, −13.21937663598824, −12.56250436845711, −12.44123320720919, −11.73538043394657, −11.38679153570114, −10.62450446868633, −10.22826065772892, −9.762192458323646, −9.287204413534933, −8.281666698190565, −8.101505681017094, −7.365112792742646, −6.908417706514205, −6.280309371402611, −5.534288448734777, −5.201800140296427, −4.508531265337160, −4.078468073318062, −3.086556881738561, −2.469976517313942, −1.790651604766325, −0.8525022806978835, 0, 0.8525022806978835, 1.790651604766325, 2.469976517313942, 3.086556881738561, 4.078468073318062, 4.508531265337160, 5.201800140296427, 5.534288448734777, 6.280309371402611, 6.908417706514205, 7.365112792742646, 8.101505681017094, 8.281666698190565, 9.287204413534933, 9.762192458323646, 10.22826065772892, 10.62450446868633, 11.38679153570114, 11.73538043394657, 12.44123320720919, 12.56250436845711, 13.21937663598824, 13.94192682504980, 14.46357932849959, 14.83480658090699

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