Properties

Label 2-8550-1.1-c1-0-138
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 + 2.35·7-s + 8-s − 4.89·11-s + 1.89·13-s + 2.35·14-s + 16-s − 2.35·17-s − 19-s − 4.89·22-s − 6.16·23-s + 1.89·26-s + 2.35·28-s − 4.27·29-s + 7.70·31-s + 32-s − 2.35·34-s − 7.78·37-s − 38-s − 3.54·41-s + 1.73·43-s − 4.89·44-s − 6.16·46-s − 5.32·47-s − 1.45·49-s + 1.89·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.889·7-s + 0.353·8-s − 1.47·11-s + 0.525·13-s + 0.629·14-s + 0.250·16-s − 0.570·17-s − 0.229·19-s − 1.04·22-s − 1.28·23-s + 0.371·26-s + 0.444·28-s − 0.793·29-s + 1.38·31-s + 0.176·32-s − 0.403·34-s − 1.28·37-s − 0.162·38-s − 0.552·41-s + 0.264·43-s − 0.737·44-s − 0.909·46-s − 0.777·47-s − 0.208·49-s + 0.262·52-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 \( 1 - 2.35T + 7T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 - 1.89T + 13T^{2} \)
17 \( 1 + 2.35T + 17T^{2} \)
23 \( 1 + 6.16T + 23T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 - 7.70T + 31T^{2} \)
37 \( 1 + 7.78T + 37T^{2} \)
41 \( 1 + 3.54T + 41T^{2} \)
43 \( 1 - 1.73T + 43T^{2} \)
47 \( 1 + 5.32T + 47T^{2} \)
53 \( 1 + 1.97T + 53T^{2} \)
59 \( 1 + 6.81T + 59T^{2} \)
61 \( 1 - 3.97T + 61T^{2} \)
67 \( 1 + 5.19T + 67T^{2} \)
71 \( 1 - 4.14T + 71T^{2} \)
73 \( 1 + 4.37T + 73T^{2} \)
79 \( 1 - 9.49T + 79T^{2} \)
83 \( 1 + 4.43T + 83T^{2} \)
89 \( 1 + 3.62T + 89T^{2} \)
97 \( 1 + 7.06T + 97T^{2} \)
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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

−7.45598199395292718004097031214, −6.64437403536975662372635464262, −5.93510110653598198001483741427, −5.25197185070482291066827672635, −4.70655597383091311843768232663, −3.99596295961969125037005694598, −3.09656774780074347514882328154, −2.25837556391202282049489462434, −1.55905565398484583550640728166, 0, 1.55905565398484583550640728166, 2.25837556391202282049489462434, 3.09656774780074347514882328154, 3.99596295961969125037005694598, 4.70655597383091311843768232663, 5.25197185070482291066827672635, 5.93510110653598198001483741427, 6.64437403536975662372635464262, 7.45598199395292718004097031214

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