Properties

Label 2-8550-1.1-c1-0-7
Degree $2$
Conductor $8550$
Sign $1$
Analytic cond. $68.2720$
Root an. cond. $8.26269$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 1.07·7-s − 8-s − 6.34·11-s − 3.41·13-s − 1.07·14-s + 16-s − 5.41·17-s + 19-s + 6.34·22-s + 6.34·23-s + 3.41·26-s + 1.07·28-s + 0.340·29-s + 1.07·31-s − 32-s + 5.41·34-s − 3.41·37-s − 38-s − 7.60·41-s + 11.1·43-s − 6.34·44-s − 6.34·46-s − 6.34·47-s − 5.83·49-s − 3.41·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.407·7-s − 0.353·8-s − 1.91·11-s − 0.948·13-s − 0.288·14-s + 0.250·16-s − 1.31·17-s + 0.229·19-s + 1.35·22-s + 1.32·23-s + 0.670·26-s + 0.203·28-s + 0.0631·29-s + 0.193·31-s − 0.176·32-s + 0.929·34-s − 0.562·37-s − 0.162·38-s − 1.18·41-s + 1.70·43-s − 0.955·44-s − 0.934·46-s − 0.924·47-s − 0.833·49-s − 0.474·52-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: no
Arithmetic: yes
Character: $\chi_{8550} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7445075585\)
\(L(\frac12)\) \(\approx\) \(0.7445075585\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.07T + 7T^{2} \)
11 \( 1 + 6.34T + 11T^{2} \)
13 \( 1 + 3.41T + 13T^{2} \)
17 \( 1 + 5.41T + 17T^{2} \)
23 \( 1 - 6.34T + 23T^{2} \)
29 \( 1 - 0.340T + 29T^{2} \)
31 \( 1 - 1.07T + 31T^{2} \)
37 \( 1 + 3.41T + 37T^{2} \)
41 \( 1 + 7.60T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + 6.34T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 0.738T + 59T^{2} \)
61 \( 1 + 2.68T + 61T^{2} \)
67 \( 1 + 2.83T + 67T^{2} \)
71 \( 1 - 2.83T + 71T^{2} \)
73 \( 1 + 6.83T + 73T^{2} \)
79 \( 1 + 1.07T + 79T^{2} \)
83 \( 1 - 0.894T + 83T^{2} \)
89 \( 1 + 6.92T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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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

−7.77916823049008533927076673520, −7.24862451452729884646877695216, −6.67199170811667325021736370539, −5.63019648636041995016179304690, −5.02521117219958121031058577034, −4.48585639664261130236534176268, −3.11902141056990383847453744543, −2.56409794199333565931121708535, −1.79581902305663366305092530818, −0.44685535770629483631864430847, 0.44685535770629483631864430847, 1.79581902305663366305092530818, 2.56409794199333565931121708535, 3.11902141056990383847453744543, 4.48585639664261130236534176268, 5.02521117219958121031058577034, 5.63019648636041995016179304690, 6.67199170811667325021736370539, 7.24862451452729884646877695216, 7.77916823049008533927076673520

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