Properties

Label 2-3549-1.1-c1-0-1
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.415·2-s + 3-s − 1.82·4-s − 4.32·5-s − 0.415·6-s − 7-s + 1.59·8-s + 9-s + 1.79·10-s − 3.32·11-s − 1.82·12-s + 0.415·14-s − 4.32·15-s + 2.99·16-s − 5.26·17-s − 0.415·18-s − 0.118·19-s + 7.89·20-s − 21-s + 1.38·22-s − 7.51·23-s + 1.59·24-s + 13.6·25-s + 27-s + 1.82·28-s − 2.46·29-s + 1.79·30-s + ⋯
L(s)  = 1  − 0.293·2-s + 0.577·3-s − 0.913·4-s − 1.93·5-s − 0.169·6-s − 0.377·7-s + 0.562·8-s + 0.333·9-s + 0.567·10-s − 1.00·11-s − 0.527·12-s + 0.111·14-s − 1.11·15-s + 0.748·16-s − 1.27·17-s − 0.0979·18-s − 0.0271·19-s + 1.76·20-s − 0.218·21-s + 0.294·22-s − 1.56·23-s + 0.324·24-s + 2.73·25-s + 0.192·27-s + 0.345·28-s − 0.456·29-s + 0.327·30-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

\(L(1)\) \(\approx\) \(0.1670602078\)
\(L(\frac12)\) \(\approx\) \(0.1670602078\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.415T + 2T^{2} \)
5 \( 1 + 4.32T + 5T^{2} \)
11 \( 1 + 3.32T + 11T^{2} \)
17 \( 1 + 5.26T + 17T^{2} \)
19 \( 1 + 0.118T + 19T^{2} \)
23 \( 1 + 7.51T + 23T^{2} \)
29 \( 1 + 2.46T + 29T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 + 5.38T + 37T^{2} \)
41 \( 1 + 11.7T + 41T^{2} \)
43 \( 1 + 0.378T + 43T^{2} \)
47 \( 1 + 3.46T + 47T^{2} \)
53 \( 1 - 4.33T + 53T^{2} \)
59 \( 1 + 8.88T + 59T^{2} \)
61 \( 1 + 6.67T + 61T^{2} \)
67 \( 1 - 3.42T + 67T^{2} \)
71 \( 1 - 7.25T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 0.278T + 79T^{2} \)
83 \( 1 - 14.5T + 83T^{2} \)
89 \( 1 - 7.54T + 89T^{2} \)
97 \( 1 + 7.20T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.527545638849920058943165674193, −7.898890815581124892202149487986, −7.47325531028596235858004827020, −6.62068523852968698899225935094, −5.24908857167690407628093466326, −4.58583913524794148980660140169, −3.74425924507920167965402549181, −3.41529670215800802482167909239, −2.01853902290070429038940148680, −0.23321418418762523549860123498, 0.23321418418762523549860123498, 2.01853902290070429038940148680, 3.41529670215800802482167909239, 3.74425924507920167965402549181, 4.58583913524794148980660140169, 5.24908857167690407628093466326, 6.62068523852968698899225935094, 7.47325531028596235858004827020, 7.898890815581124892202149487986, 8.527545638849920058943165674193

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