Properties

Label 2-54208-1.1-c1-0-40
Degree $2$
Conductor $54208$
Sign $-1$
Analytic cond. $432.853$
Root an. cond. $20.8051$
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  − 2·5-s − 7-s − 3·9-s + 2·13-s + 6·17-s − 8·19-s − 25-s + 6·29-s − 8·31-s + 2·35-s + 2·37-s − 2·41-s + 4·43-s + 6·45-s + 8·47-s + 49-s − 6·53-s − 6·61-s + 3·63-s − 4·65-s − 4·67-s + 8·71-s − 10·73-s + 16·79-s + 9·81-s − 8·83-s − 12·85-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s − 9-s + 0.554·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.338·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s + 0.894·45-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.768·61-s + 0.377·63-s − 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s + 1.80·79-s + 81-s − 0.878·83-s − 1.30·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54208\)    =    \(2^{6} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(432.853\)
Root analytic conductor: \(20.8051\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{54208} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 54208,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.78197819284230, −14.06196040134424, −13.92493435879931, −12.95855795400210, −12.65189600887553, −12.08499891078422, −11.73664834015435, −10.93849618335687, −10.81053784340531, −10.12917052057670, −9.467156377697262, −8.754338138487392, −8.557454917997878, −7.803571553528539, −7.558500425343591, −6.691254701108391, −6.128348071336211, −5.748491707837414, −5.025595484475610, −4.249010826332003, −3.773409410390329, −3.213799574604303, −2.597864764686982, −1.753446905352489, −0.7463639823046207, 0, 0.7463639823046207, 1.753446905352489, 2.597864764686982, 3.213799574604303, 3.773409410390329, 4.249010826332003, 5.025595484475610, 5.748491707837414, 6.128348071336211, 6.691254701108391, 7.558500425343591, 7.803571553528539, 8.557454917997878, 8.754338138487392, 9.467156377697262, 10.12917052057670, 10.81053784340531, 10.93849618335687, 11.73664834015435, 12.08499891078422, 12.65189600887553, 12.95855795400210, 13.92493435879931, 14.06196040134424, 14.78197819284230

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