Properties

Label 2-8550-1.1-c1-0-54
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 4.76·7-s − 8-s − 0.960·11-s + 2.24·13-s − 4.76·14-s + 16-s + 0.249·17-s + 19-s + 0.960·22-s + 9.01·23-s − 2.24·26-s + 4.76·28-s − 6.24·29-s + 2.96·31-s − 32-s − 0.249·34-s − 0.0399·37-s − 38-s − 4.96·43-s − 0.960·44-s − 9.01·46-s − 9.49·47-s + 15.7·49-s + 2.24·52-s + 6.84·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.80·7-s − 0.353·8-s − 0.289·11-s + 0.623·13-s − 1.27·14-s + 0.250·16-s + 0.0605·17-s + 0.229·19-s + 0.204·22-s + 1.88·23-s − 0.441·26-s + 0.901·28-s − 1.16·29-s + 0.531·31-s − 0.176·32-s − 0.0428·34-s − 0.00656·37-s − 0.162·38-s − 0.756·43-s − 0.144·44-s − 1.32·46-s − 1.38·47-s + 2.24·49-s + 0.311·52-s + 0.940·53-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: \(0\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.109835663\)
\(L(\frac12)\) \(\approx\) \(2.109835663\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.76T + 7T^{2} \)
11 \( 1 + 0.960T + 11T^{2} \)
13 \( 1 - 2.24T + 13T^{2} \)
17 \( 1 - 0.249T + 17T^{2} \)
23 \( 1 - 9.01T + 23T^{2} \)
29 \( 1 + 6.24T + 29T^{2} \)
31 \( 1 - 2.96T + 31T^{2} \)
37 \( 1 + 0.0399T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4.96T + 43T^{2} \)
47 \( 1 + 9.49T + 47T^{2} \)
53 \( 1 - 6.84T + 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 - 7.53T + 61T^{2} \)
67 \( 1 - 5.72T + 67T^{2} \)
71 \( 1 - 9.61T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 - 6.07T + 79T^{2} \)
83 \( 1 + 7.45T + 83T^{2} \)
89 \( 1 + 4.07T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−7.932658215027898120725384839981, −7.21008328488719579784821187264, −6.68085154898125034539412318525, −5.48457584163991225089452602843, −5.21584799228143375362707506645, −4.31759631628925177117211266726, −3.39244050325859787080489787289, −2.39928089635169478746267491379, −1.58090406644084436406961904211, −0.859179324259370373552832318397, 0.859179324259370373552832318397, 1.58090406644084436406961904211, 2.39928089635169478746267491379, 3.39244050325859787080489787289, 4.31759631628925177117211266726, 5.21584799228143375362707506645, 5.48457584163991225089452602843, 6.68085154898125034539412318525, 7.21008328488719579784821187264, 7.932658215027898120725384839981

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