Properties

Label 2-8550-1.1-c1-0-91
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 + 6·11-s + 4·14-s + 16-s + 4·17-s − 19-s + 6·22-s − 4·23-s + 4·28-s + 10·29-s − 2·31-s + 32-s + 4·34-s + 4·37-s − 38-s − 10·41-s + 12·43-s + 6·44-s − 4·46-s + 9·49-s − 6·53-s + 4·56-s + 10·58-s − 4·59-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 1.80·11-s + 1.06·14-s + 1/4·16-s + 0.970·17-s − 0.229·19-s + 1.27·22-s − 0.834·23-s + 0.755·28-s + 1.85·29-s − 0.359·31-s + 0.176·32-s + 0.685·34-s + 0.657·37-s − 0.162·38-s − 1.56·41-s + 1.82·43-s + 0.904·44-s − 0.589·46-s + 9/7·49-s − 0.824·53-s + 0.534·56-s + 1.31·58-s − 0.520·59-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

\(L(1)\) \(\approx\) \(5.113729443\)
\(L(\frac12)\) \(\approx\) \(5.113729443\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−7.82440941702371093882599632825, −6.97928784799660693830124452550, −6.31664073410883494640406913040, −5.69674485102230251419580433097, −4.83643126536051389174384294154, −4.32162586882184529844881012567, −3.70382371515602142496458478975, −2.70897349007062590009475662712, −1.64783639851140525740072906287, −1.15440653027577646177790527258, 1.15440653027577646177790527258, 1.64783639851140525740072906287, 2.70897349007062590009475662712, 3.70382371515602142496458478975, 4.32162586882184529844881012567, 4.83643126536051389174384294154, 5.69674485102230251419580433097, 6.31664073410883494640406913040, 6.97928784799660693830124452550, 7.82440941702371093882599632825

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