Properties

Label 2-8550-1.1-c1-0-78
Degree $2$
Conductor $8550$
Sign $1$
Analytic cond. $68.2720$
Root an. cond. $8.26269$
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  + 2-s + 4-s + 4·7-s + 8-s + 4·13-s + 4·14-s + 16-s + 6·17-s + 19-s − 6·23-s + 4·26-s + 4·28-s − 6·29-s + 2·31-s + 32-s + 6·34-s + 4·37-s + 38-s − 6·41-s + 4·43-s − 6·46-s + 6·47-s + 9·49-s + 4·52-s + 6·53-s + 4·56-s − 6·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 1.10·13-s + 1.06·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s − 1.25·23-s + 0.784·26-s + 0.755·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s + 1.02·34-s + 0.657·37-s + 0.162·38-s − 0.937·41-s + 0.609·43-s − 0.884·46-s + 0.875·47-s + 9/7·49-s + 0.554·52-s + 0.824·53-s + 0.534·56-s − 0.787·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8550\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(68.2720\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.780728982\)
\(L(\frac12)\) \(\approx\) \(4.780728982\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.70097791479153211453619417755, −7.22584614781181397611801863374, −6.12602123734563039032805201410, −5.61440500425845973312912145894, −5.11806274676663495258347375100, −4.09030165888843990239173890258, −3.79449350274307596703126602160, −2.67810585413014879641600772416, −1.74961074166436521935542159057, −1.06259828383013969636779705746, 1.06259828383013969636779705746, 1.74961074166436521935542159057, 2.67810585413014879641600772416, 3.79449350274307596703126602160, 4.09030165888843990239173890258, 5.11806274676663495258347375100, 5.61440500425845973312912145894, 6.12602123734563039032805201410, 7.22584614781181397611801863374, 7.70097791479153211453619417755

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