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Properties

Label 2-46200-1.1-c1-0-3
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 − 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: \(0\)
Selberg data: \((2,\ 46200,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9652283784\)
\(L(\frac12)\) \(\approx\) \(0.9652283784\)
\(L(\frac{3}{2})\) 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

−14.61277144273271, −14.10051498132981, −13.45151333248476, −13.04601795552261, −12.54287988397927, −12.14298697407114, −11.32498723474736, −11.04734323989395, −10.64464911509001, −9.798466635408741, −9.514303943241625, −8.886317989196808, −8.340361434913255, −7.502647200395755, −7.086053507842750, −6.724339246631018, −5.827237083388407, −5.430346783200375, −4.959161574922090, −4.185831099907612, −3.477057843925535, −2.986740843650360, −2.036612587619076, −1.360146592707539, −0.3704841256120259, 0.3704841256120259, 1.360146592707539, 2.036612587619076, 2.986740843650360, 3.477057843925535, 4.185831099907612, 4.959161574922090, 5.430346783200375, 5.827237083388407, 6.724339246631018, 7.086053507842750, 7.502647200395755, 8.340361434913255, 8.886317989196808, 9.514303943241625, 9.798466635408741, 10.64464911509001, 11.04734323989395, 11.32498723474736, 12.14298697407114, 12.54287988397927, 13.04601795552261, 13.45151333248476, 14.10051498132981, 14.61277144273271

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