Properties

Label 2-8550-1.1-c1-0-105
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 + 3.12·7-s − 8-s + 2·11-s − 4·13-s − 3.12·14-s + 16-s − 3.12·17-s − 19-s − 2·22-s + 4·26-s + 3.12·28-s − 2·29-s + 9.12·31-s − 32-s + 3.12·34-s + 38-s − 5.12·41-s − 10.2·43-s + 2·44-s − 10.2·47-s + 2.75·49-s − 4·52-s + 4.24·53-s − 3.12·56-s + 2·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.18·7-s − 0.353·8-s + 0.603·11-s − 1.10·13-s − 0.834·14-s + 0.250·16-s − 0.757·17-s − 0.229·19-s − 0.426·22-s + 0.784·26-s + 0.590·28-s − 0.371·29-s + 1.63·31-s − 0.176·32-s + 0.535·34-s + 0.162·38-s − 0.800·41-s − 1.56·43-s + 0.301·44-s − 1.49·47-s + 0.393·49-s − 0.554·52-s + 0.583·53-s − 0.417·56-s + 0.262·58-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 \( 1 - 3.12T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + 3.12T + 17T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 9.12T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 5.12T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 3.12T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 6.24T + 67T^{2} \)
71 \( 1 - 6.24T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 9.12T + 79T^{2} \)
83 \( 1 + 6.87T + 83T^{2} \)
89 \( 1 + 11.3T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−7.53065022155542769193869380124, −6.81855703903069503138138871376, −6.34078914137492780766939385891, −5.19136465650262397296598270495, −4.77562456258633642038820441409, −3.92329947105747106468742342945, −2.82231134717789269539524552329, −2.02033494488443966428932173692, −1.30240109854654402059735310272, 0, 1.30240109854654402059735310272, 2.02033494488443966428932173692, 2.82231134717789269539524552329, 3.92329947105747106468742342945, 4.77562456258633642038820441409, 5.19136465650262397296598270495, 6.34078914137492780766939385891, 6.81855703903069503138138871376, 7.53065022155542769193869380124

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