Properties

Label 2-3549-1.1-c1-0-15
Degree $2$
Conductor $3549$
Sign $1$
Analytic cond. $28.3389$
Root an. cond. $5.32343$
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  − 2.31·2-s − 3-s + 3.34·4-s − 1.46·5-s + 2.31·6-s − 7-s − 3.10·8-s + 9-s + 3.37·10-s − 0.0565·11-s − 3.34·12-s + 2.31·14-s + 1.46·15-s + 0.489·16-s + 5.75·17-s − 2.31·18-s + 0.807·19-s − 4.88·20-s + 21-s + 0.130·22-s + 6.01·23-s + 3.10·24-s − 2.86·25-s − 27-s − 3.34·28-s + 3.22·29-s − 3.37·30-s + ⋯
L(s)  = 1  − 1.63·2-s − 0.577·3-s + 1.67·4-s − 0.653·5-s + 0.943·6-s − 0.377·7-s − 1.09·8-s + 0.333·9-s + 1.06·10-s − 0.0170·11-s − 0.965·12-s + 0.617·14-s + 0.377·15-s + 0.122·16-s + 1.39·17-s − 0.544·18-s + 0.185·19-s − 1.09·20-s + 0.218·21-s + 0.0278·22-s + 1.25·23-s + 0.633·24-s − 0.573·25-s − 0.192·27-s − 0.631·28-s + 0.598·29-s − 0.616·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(28.3389\)
Root analytic conductor: \(5.32343\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4732543401\)
\(L(\frac12)\) \(\approx\) \(0.4732543401\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + 2.31T + 2T^{2} \)
5 \( 1 + 1.46T + 5T^{2} \)
11 \( 1 + 0.0565T + 11T^{2} \)
17 \( 1 - 5.75T + 17T^{2} \)
19 \( 1 - 0.807T + 19T^{2} \)
23 \( 1 - 6.01T + 23T^{2} \)
29 \( 1 - 3.22T + 29T^{2} \)
31 \( 1 + 5.59T + 31T^{2} \)
37 \( 1 + 0.780T + 37T^{2} \)
41 \( 1 - 5.30T + 41T^{2} \)
43 \( 1 - 2.38T + 43T^{2} \)
47 \( 1 - 3.32T + 47T^{2} \)
53 \( 1 + 9.52T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 8.88T + 61T^{2} \)
67 \( 1 + 7.80T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 13.1T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 - 7.70T + 83T^{2} \)
89 \( 1 + 8.12T + 89T^{2} \)
97 \( 1 + 3.93T + 97T^{2} \)
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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

−8.611154943375625433456528626798, −7.70077733854336352356474902584, −7.46989332109595252145442486724, −6.61407704798638433571196661428, −5.80977933853031182477361696894, −4.86882795709081720583162268952, −3.74518959578733995564712161303, −2.80008185553775102516968277993, −1.47313895437470080101977913380, −0.56994860030203061491952043006, 0.56994860030203061491952043006, 1.47313895437470080101977913380, 2.80008185553775102516968277993, 3.74518959578733995564712161303, 4.86882795709081720583162268952, 5.80977933853031182477361696894, 6.61407704798638433571196661428, 7.46989332109595252145442486724, 7.70077733854336352356474902584, 8.611154943375625433456528626798

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