Properties

Label 2-8550-1.1-c1-0-103
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 + 4.24·7-s − 8-s − 1.42·11-s − 6.91·13-s − 4.24·14-s + 16-s + 5.10·17-s + 19-s + 1.42·22-s − 3.67·23-s + 6.91·26-s + 4.24·28-s − 8.10·29-s − 1.28·31-s − 32-s − 5.10·34-s − 0.856·37-s − 38-s + 8.01·41-s + 3.57·43-s − 1.42·44-s + 3.67·46-s + 3.81·47-s + 11.0·49-s − 6.91·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.60·7-s − 0.353·8-s − 0.430·11-s − 1.91·13-s − 1.13·14-s + 0.250·16-s + 1.23·17-s + 0.229·19-s + 0.304·22-s − 0.765·23-s + 1.35·26-s + 0.802·28-s − 1.50·29-s − 0.231·31-s − 0.176·32-s − 0.874·34-s − 0.140·37-s − 0.162·38-s + 1.25·41-s + 0.544·43-s − 0.215·44-s + 0.541·46-s + 0.556·47-s + 1.57·49-s − 0.959·52-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 \( 1 - 4.24T + 7T^{2} \)
11 \( 1 + 1.42T + 11T^{2} \)
13 \( 1 + 6.91T + 13T^{2} \)
17 \( 1 - 5.10T + 17T^{2} \)
23 \( 1 + 3.67T + 23T^{2} \)
29 \( 1 + 8.10T + 29T^{2} \)
31 \( 1 + 1.28T + 31T^{2} \)
37 \( 1 + 0.856T + 37T^{2} \)
41 \( 1 - 8.01T + 41T^{2} \)
43 \( 1 - 3.57T + 43T^{2} \)
47 \( 1 - 3.81T + 47T^{2} \)
53 \( 1 + 9.06T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 8.20T + 61T^{2} \)
67 \( 1 - 4.38T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 5.38T + 73T^{2} \)
79 \( 1 - 2.14T + 79T^{2} \)
83 \( 1 + 1.04T + 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 - 6.81T + 97T^{2} \)
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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

−7.66702583766445677117171309492, −7.20661962246813719994063235744, −5.94666299093659363825244410034, −5.35082948945448919236390012031, −4.79337761188005306849948343453, −3.91642609112448143530187657434, −2.75481601160513402678995752089, −2.08699011560941518534492632333, −1.28341534067451476667875814707, 0, 1.28341534067451476667875814707, 2.08699011560941518534492632333, 2.75481601160513402678995752089, 3.91642609112448143530187657434, 4.79337761188005306849948343453, 5.35082948945448919236390012031, 5.94666299093659363825244410034, 7.20661962246813719994063235744, 7.66702583766445677117171309492

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