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Properties

Label 2-46200-1.1-c1-0-78
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 + 2·17-s + 4·19-s + 21-s − 6·23-s + 27-s − 4·29-s − 10·31-s − 33-s + 8·37-s + 6·41-s + 12·43-s − 8·47-s + 49-s + 2·51-s + 6·53-s + 4·57-s − 8·59-s + 2·61-s + 63-s − 8·67-s − 6·69-s + 8·71-s − 8·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.485·17-s + 0.917·19-s + 0.218·21-s − 1.25·23-s + 0.192·27-s − 0.742·29-s − 1.79·31-s − 0.174·33-s + 1.31·37-s + 0.937·41-s + 1.82·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 1.04·59-s + 0.256·61-s + 0.125·63-s − 0.977·67-s − 0.722·69-s + 0.949·71-s − 0.936·73-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 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
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

−14.71226123331544, −14.40681297963592, −13.95790914565834, −13.33481385810895, −12.88957594062760, −12.32235299277323, −11.86093017688422, −11.06292713025300, −10.90109709546017, −10.06713308940069, −9.499627112756931, −9.282284583994749, −8.491234936670926, −7.885666060328849, −7.560302861039921, −7.129141964468557, −6.158433126294946, −5.651554617974709, −5.217427496637840, −4.184563207626782, −4.017575445369047, −3.105049782529377, −2.553874813970012, −1.794505595196470, −1.129303489325887, 0, 1.129303489325887, 1.794505595196470, 2.553874813970012, 3.105049782529377, 4.017575445369047, 4.184563207626782, 5.217427496637840, 5.651554617974709, 6.158433126294946, 7.129141964468557, 7.560302861039921, 7.885666060328849, 8.491234936670926, 9.282284583994749, 9.499627112756931, 10.06713308940069, 10.90109709546017, 11.06292713025300, 11.86093017688422, 12.32235299277323, 12.88957594062760, 13.33481385810895, 13.95790914565834, 14.40681297963592, 14.71226123331544

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