Properties

Label 2-3549-1.1-c1-0-139
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 $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.24·2-s − 3-s − 0.445·4-s + 4.06·5-s − 1.24·6-s − 7-s − 3.04·8-s + 9-s + 5.07·10-s − 0.456·11-s + 0.445·12-s − 1.24·14-s − 4.06·15-s − 2.91·16-s − 5.72·17-s + 1.24·18-s + 0.188·19-s − 1.81·20-s + 21-s − 0.569·22-s − 5.05·23-s + 3.04·24-s + 11.5·25-s − 27-s + 0.445·28-s − 6.16·29-s − 5.07·30-s + ⋯
L(s)  = 1  + 0.881·2-s − 0.577·3-s − 0.222·4-s + 1.82·5-s − 0.509·6-s − 0.377·7-s − 1.07·8-s + 0.333·9-s + 1.60·10-s − 0.137·11-s + 0.128·12-s − 0.333·14-s − 1.05·15-s − 0.727·16-s − 1.38·17-s + 0.293·18-s + 0.0432·19-s − 0.405·20-s + 0.218·21-s − 0.121·22-s − 1.05·23-s + 0.622·24-s + 2.31·25-s − 0.192·27-s + 0.0841·28-s − 1.14·29-s − 0.926·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3549,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 + T \)
7 \( 1 + T \)
13 \( 1 \)
good2 \( 1 - 1.24T + 2T^{2} \)
5 \( 1 - 4.06T + 5T^{2} \)
11 \( 1 + 0.456T + 11T^{2} \)
17 \( 1 + 5.72T + 17T^{2} \)
19 \( 1 - 0.188T + 19T^{2} \)
23 \( 1 + 5.05T + 23T^{2} \)
29 \( 1 + 6.16T + 29T^{2} \)
31 \( 1 - 1.19T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 2.45T + 41T^{2} \)
43 \( 1 - 4.28T + 43T^{2} \)
47 \( 1 + 7.89T + 47T^{2} \)
53 \( 1 - 5.67T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 + 0.497T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 7.94T + 73T^{2} \)
79 \( 1 - 4.07T + 79T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + 9.85T + 97T^{2} \)
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

−8.368158868647085385283519093579, −6.97130388933312932882338225430, −6.40322532043637264177882285391, −5.81931819343526976400100885885, −5.27396901164943975372431744144, −4.56708094070636324918409868809, −3.60506058322161640535281476863, −2.51832252966962291920432052055, −1.71267112218039215527183958272, 0, 1.71267112218039215527183958272, 2.51832252966962291920432052055, 3.60506058322161640535281476863, 4.56708094070636324918409868809, 5.27396901164943975372431744144, 5.81931819343526976400100885885, 6.40322532043637264177882285391, 6.97130388933312932882338225430, 8.368158868647085385283519093579

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