Properties

Label 2-549-1.1-c1-0-18
Degree $2$
Conductor $549$
Sign $1$
Analytic cond. $4.38378$
Root an. cond. $2.09374$
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  + 1.97·2-s + 1.92·4-s + 3.18·5-s + 2.80·7-s − 0.158·8-s + 6.31·10-s − 3.55·11-s − 1.80·13-s + 5.54·14-s − 4.15·16-s − 4.59·17-s + 6.89·19-s + 6.12·20-s − 7.04·22-s − 5.39·23-s + 5.16·25-s − 3.57·26-s + 5.37·28-s + 4.85·29-s + 1.48·31-s − 7.90·32-s − 9.09·34-s + 8.92·35-s − 10.6·37-s + 13.6·38-s − 0.503·40-s + 7.66·41-s + ⋯
L(s)  = 1  + 1.40·2-s + 0.960·4-s + 1.42·5-s + 1.05·7-s − 0.0558·8-s + 1.99·10-s − 1.07·11-s − 0.500·13-s + 1.48·14-s − 1.03·16-s − 1.11·17-s + 1.58·19-s + 1.36·20-s − 1.50·22-s − 1.12·23-s + 1.03·25-s − 0.701·26-s + 1.01·28-s + 0.901·29-s + 0.266·31-s − 1.39·32-s − 1.56·34-s + 1.50·35-s − 1.75·37-s + 2.21·38-s − 0.0796·40-s + 1.19·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(549\)    =    \(3^{2} \cdot 61\)
Sign: $1$
Analytic conductor: \(4.38378\)
Root analytic conductor: \(2.09374\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 549,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.522507967\)
\(L(\frac12)\) \(\approx\) \(3.522507967\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
61 \( 1 + T \)
good2 \( 1 - 1.97T + 2T^{2} \)
5 \( 1 - 3.18T + 5T^{2} \)
7 \( 1 - 2.80T + 7T^{2} \)
11 \( 1 + 3.55T + 11T^{2} \)
13 \( 1 + 1.80T + 13T^{2} \)
17 \( 1 + 4.59T + 17T^{2} \)
19 \( 1 - 6.89T + 19T^{2} \)
23 \( 1 + 5.39T + 23T^{2} \)
29 \( 1 - 4.85T + 29T^{2} \)
31 \( 1 - 1.48T + 31T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 - 7.66T + 41T^{2} \)
43 \( 1 - 6.99T + 43T^{2} \)
47 \( 1 + 2.57T + 47T^{2} \)
53 \( 1 + 5.33T + 53T^{2} \)
59 \( 1 - 7.29T + 59T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 - 6.15T + 73T^{2} \)
79 \( 1 + 0.700T + 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + 7.93T + 89T^{2} \)
97 \( 1 + 4.27T + 97T^{2} \)
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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

−10.95235595957302893024965742930, −10.05598747907515054108445981468, −9.147682400746192332686831279330, −7.991275360541692099475837946418, −6.84331521380531412818945972756, −5.75002566909106858928358953105, −5.22517069300163741598722515620, −4.44342547469520007809416202157, −2.84586198646766296517158255473, −1.97042961572849279502382844784, 1.97042961572849279502382844784, 2.84586198646766296517158255473, 4.44342547469520007809416202157, 5.22517069300163741598722515620, 5.75002566909106858928358953105, 6.84331521380531412818945972756, 7.991275360541692099475837946418, 9.147682400746192332686831279330, 10.05598747907515054108445981468, 10.95235595957302893024965742930

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