Properties

Label 2-8550-1.1-c1-0-133
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 − 1.03·7-s + 8-s + 3.86·11-s + 1.03·13-s − 1.03·14-s + 16-s − 7.46·17-s − 19-s + 3.86·22-s + 1.46·23-s + 1.03·26-s − 1.03·28-s − 9.52·29-s + 2.92·31-s + 32-s − 7.46·34-s − 6.69·37-s − 38-s − 6.69·41-s − 1.79·43-s + 3.86·44-s + 1.46·46-s − 9.46·47-s − 5.92·49-s + 1.03·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.391·7-s + 0.353·8-s + 1.16·11-s + 0.287·13-s − 0.276·14-s + 0.250·16-s − 1.81·17-s − 0.229·19-s + 0.823·22-s + 0.305·23-s + 0.203·26-s − 0.195·28-s − 1.76·29-s + 0.525·31-s + 0.176·32-s − 1.28·34-s − 1.10·37-s − 0.162·38-s − 1.04·41-s − 0.273·43-s + 0.582·44-s + 0.215·46-s − 1.38·47-s − 0.846·49-s + 0.143·52-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 \( 1 + 1.03T + 7T^{2} \)
11 \( 1 - 3.86T + 11T^{2} \)
13 \( 1 - 1.03T + 13T^{2} \)
17 \( 1 + 7.46T + 17T^{2} \)
23 \( 1 - 1.46T + 23T^{2} \)
29 \( 1 + 9.52T + 29T^{2} \)
31 \( 1 - 2.92T + 31T^{2} \)
37 \( 1 + 6.69T + 37T^{2} \)
41 \( 1 + 6.69T + 41T^{2} \)
43 \( 1 + 1.79T + 43T^{2} \)
47 \( 1 + 9.46T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 3.58T + 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 + 13.3T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 4.39T + 83T^{2} \)
89 \( 1 + 1.03T + 89T^{2} \)
97 \( 1 - 17.2T + 97T^{2} \)
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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

−7.06803821910788384113658907932, −6.67045166744698232027563618257, −6.21144673719911554374151535173, −5.26636700491076218577964732557, −4.59711051783282614574722854794, −3.81895567213535687052661989691, −3.33949799656411431151792646017, −2.21257853026140678874406563119, −1.52380750299865813559141647063, 0, 1.52380750299865813559141647063, 2.21257853026140678874406563119, 3.33949799656411431151792646017, 3.81895567213535687052661989691, 4.59711051783282614574722854794, 5.26636700491076218577964732557, 6.21144673719911554374151535173, 6.67045166744698232027563618257, 7.06803821910788384113658907932

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