Properties

Label 2-8550-1.1-c1-0-101
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 + 4.44·7-s − 8-s − 3.44·11-s − 2.44·13-s − 4.44·14-s + 16-s − 4.44·17-s + 19-s + 3.44·22-s + 23-s + 2.44·26-s + 4.44·28-s + 4.34·29-s − 3·31-s − 32-s + 4.44·34-s − 7.79·37-s − 38-s + 0.898·41-s − 2.44·43-s − 3.44·44-s − 46-s + 7.79·47-s + 12.7·49-s − 2.44·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.68·7-s − 0.353·8-s − 1.04·11-s − 0.679·13-s − 1.18·14-s + 0.250·16-s − 1.07·17-s + 0.229·19-s + 0.735·22-s + 0.208·23-s + 0.480·26-s + 0.840·28-s + 0.807·29-s − 0.538·31-s − 0.176·32-s + 0.763·34-s − 1.28·37-s − 0.162·38-s + 0.140·41-s − 0.373·43-s − 0.520·44-s − 0.147·46-s + 1.13·47-s + 1.82·49-s − 0.339·52-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 \( 1 - 4.44T + 7T^{2} \)
11 \( 1 + 3.44T + 11T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
23 \( 1 - T + 23T^{2} \)
29 \( 1 - 4.34T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 7.79T + 37T^{2} \)
41 \( 1 - 0.898T + 41T^{2} \)
43 \( 1 + 2.44T + 43T^{2} \)
47 \( 1 - 7.79T + 47T^{2} \)
53 \( 1 - 7.44T + 53T^{2} \)
59 \( 1 + 6.44T + 59T^{2} \)
61 \( 1 + 9.44T + 61T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 - 1.55T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 8.34T + 83T^{2} \)
89 \( 1 + 2.10T + 89T^{2} \)
97 \( 1 - 1.55T + 97T^{2} \)
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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

−7.39163410571910708312183102178, −7.17545928201502688642327670180, −6.09293617390758347831210042054, −5.15163857012564456016355525999, −4.89863233991983043494807823760, −3.93957184210039296949266937898, −2.68922285643142487106369187184, −2.15073039618582591710926159104, −1.26266098821697939530861211072, 0, 1.26266098821697939530861211072, 2.15073039618582591710926159104, 2.68922285643142487106369187184, 3.93957184210039296949266937898, 4.89863233991983043494807823760, 5.15163857012564456016355525999, 6.09293617390758347831210042054, 7.17545928201502688642327670180, 7.39163410571910708312183102178

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