Properties

Label 2-8550-1.1-c1-0-14
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 + 2.46·7-s − 8-s − 0.728·11-s − 6.23·13-s − 2.46·14-s + 16-s − 0.563·17-s − 19-s + 0.728·22-s − 4.63·23-s + 6.23·26-s + 2.46·28-s − 10.2·29-s + 6.06·31-s − 32-s + 0.563·34-s − 5.72·37-s + 38-s − 4.79·41-s + 8.06·43-s − 0.728·44-s + 4.63·46-s − 8.12·47-s − 0.900·49-s − 6.23·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.933·7-s − 0.353·8-s − 0.219·11-s − 1.72·13-s − 0.660·14-s + 0.250·16-s − 0.136·17-s − 0.229·19-s + 0.155·22-s − 0.966·23-s + 1.22·26-s + 0.466·28-s − 1.89·29-s + 1.08·31-s − 0.176·32-s + 0.0965·34-s − 0.941·37-s + 0.162·38-s − 0.748·41-s + 1.23·43-s − 0.109·44-s + 0.683·46-s − 1.18·47-s − 0.128·49-s − 0.864·52-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: \(0\)
Selberg data: \((2,\ 8550,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.039860584\)
\(L(\frac12)\) \(\approx\) \(1.039860584\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.46T + 7T^{2} \)
11 \( 1 + 0.728T + 11T^{2} \)
13 \( 1 + 6.23T + 13T^{2} \)
17 \( 1 + 0.563T + 17T^{2} \)
23 \( 1 + 4.63T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 - 6.06T + 31T^{2} \)
37 \( 1 + 5.72T + 37T^{2} \)
41 \( 1 + 4.79T + 41T^{2} \)
43 \( 1 - 8.06T + 43T^{2} \)
47 \( 1 + 8.12T + 47T^{2} \)
53 \( 1 - 1.53T + 53T^{2} \)
59 \( 1 - 5.76T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 - 4.39T + 71T^{2} \)
73 \( 1 + 4.09T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + 7.85T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−7.975239392201194734833546528512, −7.20711779628335621987261197324, −6.67131918531023650799606680141, −5.62402299051102079954756523583, −5.09002319492617793987502708246, −4.33982975164361804492220988575, −3.38425539607140498469565534189, −2.21501035721447891472946580252, −1.94696767510668984163891284021, −0.53093777693241158996295069653, 0.53093777693241158996295069653, 1.94696767510668984163891284021, 2.21501035721447891472946580252, 3.38425539607140498469565534189, 4.33982975164361804492220988575, 5.09002319492617793987502708246, 5.62402299051102079954756523583, 6.67131918531023650799606680141, 7.20711779628335621987261197324, 7.975239392201194734833546528512

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