Properties

Label 2-8550-1.1-c1-0-115
Degree $2$
Conductor $8550$
Sign $-1$
Analytic cond. $68.2720$
Root an. cond. $8.26269$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 2·7-s − 8-s − 4·11-s + 6·13-s − 2·14-s + 16-s + 4·17-s + 19-s + 4·22-s + 4·23-s − 6·26-s + 2·28-s − 6·29-s − 6·31-s − 32-s − 4·34-s − 10·37-s − 38-s − 4·41-s − 12·43-s − 4·44-s − 4·46-s + 4·47-s − 3·49-s + 6·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 1.20·11-s + 1.66·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s + 0.229·19-s + 0.852·22-s + 0.834·23-s − 1.17·26-s + 0.377·28-s − 1.11·29-s − 1.07·31-s − 0.176·32-s − 0.685·34-s − 1.64·37-s − 0.162·38-s − 0.624·41-s − 1.82·43-s − 0.603·44-s − 0.589·46-s + 0.583·47-s − 3/7·49-s + 0.832·52-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: yes
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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.69787464173758314207740152450, −6.92465975204620545886021919982, −6.06308608778305379850346973105, −5.37817775904103702307919769295, −4.86802437783647759141855172545, −3.53523660403903799144368606175, −3.19145578399769463338290811972, −1.85387130165108636351727613914, −1.35376053111283218101424788308, 0, 1.35376053111283218101424788308, 1.85387130165108636351727613914, 3.19145578399769463338290811972, 3.53523660403903799144368606175, 4.86802437783647759141855172545, 5.37817775904103702307919769295, 6.06308608778305379850346973105, 6.92465975204620545886021919982, 7.69787464173758314207740152450

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