Properties

Label 2-8550-1.1-c1-0-137
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 + 0.449·7-s + 8-s + 1.44·11-s − 2.44·13-s + 0.449·14-s + 16-s − 0.449·17-s + 19-s + 1.44·22-s − 23-s − 2.44·26-s + 0.449·28-s − 10.3·29-s − 3·31-s + 32-s − 0.449·34-s − 11.7·37-s + 38-s − 8.89·41-s − 2.44·43-s + 1.44·44-s − 46-s + 11.7·47-s − 6.79·49-s − 2.44·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.169·7-s + 0.353·8-s + 0.437·11-s − 0.679·13-s + 0.120·14-s + 0.250·16-s − 0.109·17-s + 0.229·19-s + 0.309·22-s − 0.208·23-s − 0.480·26-s + 0.0849·28-s − 1.92·29-s − 0.538·31-s + 0.176·32-s − 0.0770·34-s − 1.93·37-s + 0.162·38-s − 1.38·41-s − 0.373·43-s + 0.218·44-s − 0.147·46-s + 1.72·47-s − 0.971·49-s − 0.339·52-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 \( 1 - 0.449T + 7T^{2} \)
11 \( 1 - 1.44T + 11T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 + 0.449T + 17T^{2} \)
23 \( 1 + T + 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 11.7T + 37T^{2} \)
41 \( 1 + 8.89T + 41T^{2} \)
43 \( 1 + 2.44T + 43T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + 2.55T + 53T^{2} \)
59 \( 1 + 1.55T + 59T^{2} \)
61 \( 1 + 4.55T + 61T^{2} \)
67 \( 1 - 9.24T + 67T^{2} \)
71 \( 1 - 6.44T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 6.34T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 6.44T + 97T^{2} \)
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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

−7.17830825071997293580752081991, −6.88653657066453332015227324422, −5.87909368744776991052025743866, −5.33749346424895015059825984495, −4.69300327518379773062357550762, −3.80219284658485546346157179261, −3.30504852747576335639947278577, −2.19354615906413294404083325074, −1.55588384700987814873690953343, 0, 1.55588384700987814873690953343, 2.19354615906413294404083325074, 3.30504852747576335639947278577, 3.80219284658485546346157179261, 4.69300327518379773062357550762, 5.33749346424895015059825984495, 5.87909368744776991052025743866, 6.88653657066453332015227324422, 7.17830825071997293580752081991

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