Properties

Label 2-8550-1.1-c1-0-13
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 − 0.0778·7-s − 8-s + 4.50·11-s − 5.33·13-s + 0.0778·14-s + 16-s − 7.33·17-s + 19-s − 4.50·22-s − 3.40·23-s + 5.33·26-s − 0.0778·28-s + 1.33·29-s − 2.50·31-s − 32-s + 7.33·34-s − 5.50·37-s − 38-s + 0.506·43-s + 4.50·44-s + 3.40·46-s + 5.66·47-s − 6.99·49-s − 5.33·52-s + 12.9·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.0294·7-s − 0.353·8-s + 1.35·11-s − 1.47·13-s + 0.0208·14-s + 0.250·16-s − 1.77·17-s + 0.229·19-s − 0.960·22-s − 0.710·23-s + 1.04·26-s − 0.0147·28-s + 0.247·29-s − 0.450·31-s − 0.176·32-s + 1.25·34-s − 0.905·37-s − 0.162·38-s + 0.0772·43-s + 0.679·44-s + 0.502·46-s + 0.825·47-s − 0.999·49-s − 0.739·52-s + 1.77·53-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

\(L(1)\) \(\approx\) \(1.007954156\)
\(L(\frac12)\) \(\approx\) \(1.007954156\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.0778T + 7T^{2} \)
11 \( 1 - 4.50T + 11T^{2} \)
13 \( 1 + 5.33T + 13T^{2} \)
17 \( 1 + 7.33T + 17T^{2} \)
23 \( 1 + 3.40T + 23T^{2} \)
29 \( 1 - 1.33T + 29T^{2} \)
31 \( 1 + 2.50T + 31T^{2} \)
37 \( 1 + 5.50T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 0.506T + 43T^{2} \)
47 \( 1 - 5.66T + 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 + 7.56T + 59T^{2} \)
61 \( 1 + 2.15T + 61T^{2} \)
67 \( 1 + 4.58T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 5.09T + 73T^{2} \)
79 \( 1 - 17.0T + 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 - 7.67T + 97T^{2} \)
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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

−7.73759807723643130438037047272, −7.07145964362155827568576902396, −6.64262732052179271302170045102, −5.92012973995622319636226417224, −4.92578600689776400714057995082, −4.27559720964908123676558172096, −3.43173422872538939244509764568, −2.34723620830707662855278933327, −1.81744556798336408316517956451, −0.52944314001153072993704818918, 0.52944314001153072993704818918, 1.81744556798336408316517956451, 2.34723620830707662855278933327, 3.43173422872538939244509764568, 4.27559720964908123676558172096, 4.92578600689776400714057995082, 5.92012973995622319636226417224, 6.64262732052179271302170045102, 7.07145964362155827568576902396, 7.73759807723643130438037047272

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