Properties

Label 2-8550-1.1-c1-0-110
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 4.12·7-s + 8-s + 2.64·11-s − 2.51·13-s − 4.12·14-s + 16-s − 0.515·17-s + 19-s + 2.64·22-s + 3.09·23-s − 2.51·26-s − 4.12·28-s + 7.79·29-s + 3.67·31-s + 32-s − 0.515·34-s − 10.2·37-s + 38-s − 8.88·41-s − 8.64·43-s + 2.64·44-s + 3.09·46-s − 4.96·47-s + 10.0·49-s − 2.51·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.55·7-s + 0.353·8-s + 0.795·11-s − 0.697·13-s − 1.10·14-s + 0.250·16-s − 0.124·17-s + 0.229·19-s + 0.562·22-s + 0.645·23-s − 0.493·26-s − 0.779·28-s + 1.44·29-s + 0.659·31-s + 0.176·32-s − 0.0883·34-s − 1.68·37-s + 0.162·38-s − 1.38·41-s − 1.31·43-s + 0.397·44-s + 0.456·46-s − 0.724·47-s + 1.43·49-s − 0.348·52-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: \(1\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.12T + 7T^{2} \)
11 \( 1 - 2.64T + 11T^{2} \)
13 \( 1 + 2.51T + 13T^{2} \)
17 \( 1 + 0.515T + 17T^{2} \)
23 \( 1 - 3.09T + 23T^{2} \)
29 \( 1 - 7.79T + 29T^{2} \)
31 \( 1 - 3.67T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 + 8.88T + 41T^{2} \)
43 \( 1 + 8.64T + 43T^{2} \)
47 \( 1 + 4.96T + 47T^{2} \)
53 \( 1 - 5.48T + 53T^{2} \)
59 \( 1 + 3.15T + 59T^{2} \)
61 \( 1 + 12.6T + 61T^{2} \)
67 \( 1 - 7.40T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 2.70T + 73T^{2} \)
79 \( 1 - 16.7T + 79T^{2} \)
83 \( 1 - 3.28T + 83T^{2} \)
89 \( 1 - 7.60T + 89T^{2} \)
97 \( 1 + 3.93T + 97T^{2} \)
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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

−7.06385191199383989682600033643, −6.61101785681870698963705330502, −6.30413941851520337767748445547, −5.20645198306941852159809004992, −4.73844675697588383476504646858, −3.67181168565848924557737223968, −3.25335353565895194655848022040, −2.49968479049327546957286625199, −1.33756216443445124111783273598, 0, 1.33756216443445124111783273598, 2.49968479049327546957286625199, 3.25335353565895194655848022040, 3.67181168565848924557737223968, 4.73844675697588383476504646858, 5.20645198306941852159809004992, 6.30413941851520337767748445547, 6.61101785681870698963705330502, 7.06385191199383989682600033643

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