Properties

Degree 2
Conductor $ 3^{2} \cdot 7 \cdot 11 \cdot 17^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 2·5-s − 7-s − 3·8-s − 2·10-s − 11-s + 6·13-s − 14-s − 16-s + 4·19-s + 2·20-s − 22-s − 25-s + 6·26-s + 28-s − 2·29-s − 8·31-s + 5·32-s + 2·35-s − 6·37-s + 4·38-s + 6·40-s + 10·41-s − 4·43-s + 44-s + 8·47-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.894·5-s − 0.377·7-s − 1.06·8-s − 0.632·10-s − 0.301·11-s + 1.66·13-s − 0.267·14-s − 1/4·16-s + 0.917·19-s + 0.447·20-s − 0.213·22-s − 1/5·25-s + 1.17·26-s + 0.188·28-s − 0.371·29-s − 1.43·31-s + 0.883·32-s + 0.338·35-s − 0.986·37-s + 0.648·38-s + 0.948·40-s + 1.56·41-s − 0.609·43-s + 0.150·44-s + 1.16·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 200277 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200277 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(200277\)    =    \(3^{2} \cdot 7 \cdot 11 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{200277} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 200277,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.107255635\)
\(L(\frac12)\)  \(\approx\)  \(1.107255635\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;7,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;7,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
17 \( 1 \)
good2 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−13.13075207342225, −12.61605690895245, −12.30413188537128, −11.61735581771380, −11.37688984272274, −10.82860633463669, −10.36641643970136, −9.617507262556442, −9.280742486550687, −8.755699937697509, −8.348233786553634, −7.860581140925973, −7.212669077751933, −6.904885759206510, −5.921704321467044, −5.818093333890081, −5.358379596197051, −4.549406596632455, −4.079277623276350, −3.748236571637200, −3.216748974656410, −2.840074665610275, −1.794169438397671, −1.097394697976248, −0.2956978177171406, 0.2956978177171406, 1.097394697976248, 1.794169438397671, 2.840074665610275, 3.216748974656410, 3.748236571637200, 4.079277623276350, 4.549406596632455, 5.358379596197051, 5.818093333890081, 5.921704321467044, 6.904885759206510, 7.212669077751933, 7.860581140925973, 8.348233786553634, 8.755699937697509, 9.280742486550687, 9.617507262556442, 10.36641643970136, 10.82860633463669, 11.37688984272274, 11.61735581771380, 12.30413188537128, 12.61605690895245, 13.13075207342225

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