Properties

Label 2-8550-1.1-c1-0-90
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 − 4.42·7-s − 8-s + 5.80·11-s + 6.42·13-s + 4.42·14-s + 16-s − 3.37·17-s − 19-s − 5.80·22-s − 6.42·23-s − 6.42·26-s − 4.42·28-s − 7.80·29-s + 9.05·31-s − 32-s + 3.37·34-s − 3.67·37-s + 38-s − 4.42·41-s + 1.05·43-s + 5.80·44-s + 6.42·46-s + 5.18·47-s + 12.6·49-s + 6.42·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.67·7-s − 0.353·8-s + 1.75·11-s + 1.78·13-s + 1.18·14-s + 0.250·16-s − 0.819·17-s − 0.229·19-s − 1.23·22-s − 1.34·23-s − 1.26·26-s − 0.836·28-s − 1.44·29-s + 1.62·31-s − 0.176·32-s + 0.579·34-s − 0.603·37-s + 0.162·38-s − 0.691·41-s + 0.160·43-s + 0.875·44-s + 0.947·46-s + 0.756·47-s + 1.80·49-s + 0.891·52-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 \( 1 + 4.42T + 7T^{2} \)
11 \( 1 - 5.80T + 11T^{2} \)
13 \( 1 - 6.42T + 13T^{2} \)
17 \( 1 + 3.37T + 17T^{2} \)
23 \( 1 + 6.42T + 23T^{2} \)
29 \( 1 + 7.80T + 29T^{2} \)
31 \( 1 - 9.05T + 31T^{2} \)
37 \( 1 + 3.67T + 37T^{2} \)
41 \( 1 + 4.42T + 41T^{2} \)
43 \( 1 - 1.05T + 43T^{2} \)
47 \( 1 - 5.18T + 47T^{2} \)
53 \( 1 + 4.75T + 53T^{2} \)
59 \( 1 + 4.62T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 2.75T + 67T^{2} \)
71 \( 1 + 7.61T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 2.94T + 79T^{2} \)
83 \( 1 + 0.133T + 83T^{2} \)
89 \( 1 - 3.18T + 89T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−7.34259352230170570617543730828, −6.60877166144834602510749654782, −6.21470719207236599319447571225, −5.90933538391036397910953443821, −4.30933140601898424930468949964, −3.73965198960801103894794851466, −3.19552415553085885307725142898, −2.01454928092403901511776494726, −1.14527809534458174151834125722, 0, 1.14527809534458174151834125722, 2.01454928092403901511776494726, 3.19552415553085885307725142898, 3.73965198960801103894794851466, 4.30933140601898424930468949964, 5.90933538391036397910953443821, 6.21470719207236599319447571225, 6.60877166144834602510749654782, 7.34259352230170570617543730828

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