Properties

Label 2-8550-1.1-c1-0-132
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 − 0.193·7-s + 8-s + 0.514·11-s − 3.12·13-s − 0.193·14-s + 16-s + 2.83·17-s + 19-s + 0.514·22-s − 2.32·23-s − 3.12·26-s − 0.193·28-s − 0.164·29-s − 9.05·31-s + 32-s + 2.83·34-s − 3.02·37-s + 38-s − 9.96·41-s − 5.51·43-s + 0.514·44-s − 2.32·46-s − 1.70·47-s − 6.96·49-s − 3.12·52-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.0730·7-s + 0.353·8-s + 0.155·11-s − 0.867·13-s − 0.0516·14-s + 0.250·16-s + 0.687·17-s + 0.229·19-s + 0.109·22-s − 0.483·23-s − 0.613·26-s − 0.0365·28-s − 0.0306·29-s − 1.62·31-s + 0.176·32-s + 0.486·34-s − 0.497·37-s + 0.162·38-s − 1.55·41-s − 0.840·43-s + 0.0775·44-s − 0.342·46-s − 0.249·47-s − 0.994·49-s − 0.433·52-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 \( 1 + 0.193T + 7T^{2} \)
11 \( 1 - 0.514T + 11T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 - 2.83T + 17T^{2} \)
23 \( 1 + 2.32T + 23T^{2} \)
29 \( 1 + 0.164T + 29T^{2} \)
31 \( 1 + 9.05T + 31T^{2} \)
37 \( 1 + 3.02T + 37T^{2} \)
41 \( 1 + 9.96T + 41T^{2} \)
43 \( 1 + 5.51T + 43T^{2} \)
47 \( 1 + 1.70T + 47T^{2} \)
53 \( 1 - 2.90T + 53T^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 - 5.92T + 61T^{2} \)
67 \( 1 + 4.22T + 67T^{2} \)
71 \( 1 + 3.16T + 71T^{2} \)
73 \( 1 - 8.37T + 73T^{2} \)
79 \( 1 - 6.02T + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 - 9.01T + 89T^{2} \)
97 \( 1 - 8.89T + 97T^{2} \)
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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

−7.39632837848852876879930681800, −6.67639189454609935946124162125, −6.02721452491203185840066350506, −5.14406886455350205191750309987, −4.86666834561171968793557830264, −3.69716321751102060155838167055, −3.33185719055502394665719115404, −2.26457949940225757702892949909, −1.50099055647686889364296435066, 0, 1.50099055647686889364296435066, 2.26457949940225757702892949909, 3.33185719055502394665719115404, 3.69716321751102060155838167055, 4.86666834561171968793557830264, 5.14406886455350205191750309987, 6.02721452491203185840066350506, 6.67639189454609935946124162125, 7.39632837848852876879930681800

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