Properties

Label 2-8550-1.1-c1-0-114
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 − 3.35·7-s + 8-s + 0.962·11-s − 1.61·13-s − 3.35·14-s + 16-s + 0.387·17-s + 19-s + 0.962·22-s + 0.962·23-s − 1.61·26-s − 3.35·28-s − 6.96·29-s + 3.35·31-s + 32-s + 0.387·34-s − 1.61·37-s + 38-s + 9.27·41-s + 6.18·43-s + 0.962·44-s + 0.962·46-s − 0.962·47-s + 4.22·49-s − 1.61·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.26·7-s + 0.353·8-s + 0.290·11-s − 0.447·13-s − 0.895·14-s + 0.250·16-s + 0.0940·17-s + 0.229·19-s + 0.205·22-s + 0.200·23-s − 0.316·26-s − 0.633·28-s − 1.29·29-s + 0.601·31-s + 0.176·32-s + 0.0665·34-s − 0.265·37-s + 0.162·38-s + 1.44·41-s + 0.943·43-s + 0.145·44-s + 0.141·46-s − 0.140·47-s + 0.603·49-s − 0.223·52-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 \( 1 + 3.35T + 7T^{2} \)
11 \( 1 - 0.962T + 11T^{2} \)
13 \( 1 + 1.61T + 13T^{2} \)
17 \( 1 - 0.387T + 17T^{2} \)
23 \( 1 - 0.962T + 23T^{2} \)
29 \( 1 + 6.96T + 29T^{2} \)
31 \( 1 - 3.35T + 31T^{2} \)
37 \( 1 + 1.61T + 37T^{2} \)
41 \( 1 - 9.27T + 41T^{2} \)
43 \( 1 - 6.18T + 43T^{2} \)
47 \( 1 + 0.962T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 7.22T + 67T^{2} \)
71 \( 1 + 7.22T + 71T^{2} \)
73 \( 1 + 3.22T + 73T^{2} \)
79 \( 1 + 3.35T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + 4.64T + 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−7.26797251122725166331103125536, −6.68659734201468240565283234543, −5.97143893759127036427898969016, −5.50325262314081907050185871832, −4.50339253326876643345873624053, −3.89957349546523834065971345229, −3.08883234970793877900312336878, −2.53408084748291412134699387442, −1.34820802244291330751383555986, 0, 1.34820802244291330751383555986, 2.53408084748291412134699387442, 3.08883234970793877900312336878, 3.89957349546523834065971345229, 4.50339253326876643345873624053, 5.50325262314081907050185871832, 5.97143893759127036427898969016, 6.68659734201468240565283234543, 7.26797251122725166331103125536

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