Properties

Label 2-8550-1.1-c1-0-123
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 − 1.47·7-s + 8-s + 2.29·11-s − 5.29·13-s − 1.47·14-s + 16-s + 1.47·17-s − 19-s + 2.29·22-s − 1.86·23-s − 5.29·26-s − 1.47·28-s − 7.16·29-s + 0.0470·31-s + 32-s + 1.47·34-s + 6.59·37-s − 38-s − 0.179·41-s + 7.98·43-s + 2.29·44-s − 1.86·46-s + 12.4·47-s − 4.82·49-s − 5.29·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.558·7-s + 0.353·8-s + 0.692·11-s − 1.46·13-s − 0.394·14-s + 0.250·16-s + 0.358·17-s − 0.229·19-s + 0.489·22-s − 0.389·23-s − 1.03·26-s − 0.279·28-s − 1.33·29-s + 0.00845·31-s + 0.176·32-s + 0.253·34-s + 1.08·37-s − 0.162·38-s − 0.0280·41-s + 1.21·43-s + 0.346·44-s − 0.275·46-s + 1.81·47-s − 0.688·49-s − 0.734·52-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 \( 1 + 1.47T + 7T^{2} \)
11 \( 1 - 2.29T + 11T^{2} \)
13 \( 1 + 5.29T + 13T^{2} \)
17 \( 1 - 1.47T + 17T^{2} \)
23 \( 1 + 1.86T + 23T^{2} \)
29 \( 1 + 7.16T + 29T^{2} \)
31 \( 1 - 0.0470T + 31T^{2} \)
37 \( 1 - 6.59T + 37T^{2} \)
41 \( 1 + 0.179T + 41T^{2} \)
43 \( 1 - 7.98T + 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + 6.34T + 59T^{2} \)
61 \( 1 + 9.93T + 61T^{2} \)
67 \( 1 + 14.8T + 67T^{2} \)
71 \( 1 + 14.0T + 71T^{2} \)
73 \( 1 + 14.4T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 - 6.11T + 83T^{2} \)
89 \( 1 - 6.46T + 89T^{2} \)
97 \( 1 - 4.42T + 97T^{2} \)
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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

−7.50205758102059509805304124407, −6.68863249250605218438765759024, −5.90693115706123916205285929007, −5.50420412663478528617957155201, −4.38674743721012348506771336811, −4.10979918409140993415180965066, −3.03129174503983615466834982550, −2.46255843408285878057994504854, −1.41367915655116801823959000825, 0, 1.41367915655116801823959000825, 2.46255843408285878057994504854, 3.03129174503983615466834982550, 4.10979918409140993415180965066, 4.38674743721012348506771336811, 5.50420412663478528617957155201, 5.90693115706123916205285929007, 6.68863249250605218438765759024, 7.50205758102059509805304124407

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