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Properties

Label 2-46200-1.1-c1-0-55
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 + 4·13-s − 4·17-s − 2·19-s − 21-s − 6·23-s − 27-s − 2·29-s − 2·31-s − 33-s + 2·37-s − 4·39-s − 12·41-s + 5·43-s − 3·47-s + 49-s + 4·51-s + 7·53-s + 2·57-s + 6·59-s + 14·61-s + 63-s + 6·69-s + 12·71-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.970·17-s − 0.458·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s − 0.371·29-s − 0.359·31-s − 0.174·33-s + 0.328·37-s − 0.640·39-s − 1.87·41-s + 0.762·43-s − 0.437·47-s + 1/7·49-s + 0.560·51-s + 0.961·53-s + 0.264·57-s + 0.781·59-s + 1.79·61-s + 0.125·63-s + 0.722·69-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 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.86639924103758, −14.38200550017022, −13.75920075244121, −13.29870532416517, −12.88976142820719, −12.19960602579615, −11.69994397512518, −11.19333762614393, −10.94393954616055, −10.14550522564499, −9.848376825196743, −8.991319525409143, −8.492178588799010, −8.185180441830897, −7.289293387353097, −6.792842209261542, −6.281715695147439, −5.704413657172641, −5.209087269375391, −4.346511040234216, −4.012139606743685, −3.366397038433647, −2.260445267531671, −1.802509319116995, −0.9336117007717372, 0, 0.9336117007717372, 1.802509319116995, 2.260445267531671, 3.366397038433647, 4.012139606743685, 4.346511040234216, 5.209087269375391, 5.704413657172641, 6.281715695147439, 6.792842209261542, 7.289293387353097, 8.185180441830897, 8.492178588799010, 8.991319525409143, 9.848376825196743, 10.14550522564499, 10.94393954616055, 11.19333762614393, 11.69994397512518, 12.19960602579615, 12.88976142820719, 13.29870532416517, 13.75920075244121, 14.38200550017022, 14.86639924103758

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