Properties

Label 2-3549-1.1-c1-0-131
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  + 0.106·2-s + 3-s − 1.98·4-s + 1.41·5-s + 0.106·6-s − 7-s − 0.424·8-s + 9-s + 0.150·10-s + 3.00·11-s − 1.98·12-s − 0.106·14-s + 1.41·15-s + 3.93·16-s − 6.36·17-s + 0.106·18-s − 5.30·19-s − 2.80·20-s − 21-s + 0.319·22-s − 0.671·23-s − 0.424·24-s − 3.00·25-s + 27-s + 1.98·28-s + 0.516·29-s + 0.150·30-s + ⋯
L(s)  = 1  + 0.0752·2-s + 0.577·3-s − 0.994·4-s + 0.631·5-s + 0.0434·6-s − 0.377·7-s − 0.149·8-s + 0.333·9-s + 0.0475·10-s + 0.904·11-s − 0.574·12-s − 0.0284·14-s + 0.364·15-s + 0.983·16-s − 1.54·17-s + 0.0250·18-s − 1.21·19-s − 0.628·20-s − 0.218·21-s + 0.0680·22-s − 0.140·23-s − 0.0865·24-s − 0.600·25-s + 0.192·27-s + 0.375·28-s + 0.0959·29-s + 0.0274·30-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 \( 1 - 0.106T + 2T^{2} \)
5 \( 1 - 1.41T + 5T^{2} \)
11 \( 1 - 3.00T + 11T^{2} \)
17 \( 1 + 6.36T + 17T^{2} \)
19 \( 1 + 5.30T + 19T^{2} \)
23 \( 1 + 0.671T + 23T^{2} \)
29 \( 1 - 0.516T + 29T^{2} \)
31 \( 1 - 0.282T + 31T^{2} \)
37 \( 1 + 6.45T + 37T^{2} \)
41 \( 1 + 3.97T + 41T^{2} \)
43 \( 1 - 4.94T + 43T^{2} \)
47 \( 1 - 5.91T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 2.30T + 59T^{2} \)
61 \( 1 - 4.03T + 61T^{2} \)
67 \( 1 - 2.11T + 67T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + 6.44T + 73T^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 - 8.49T + 83T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + 0.366T + 97T^{2} \)
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

−8.533066462293558207418474625831, −7.53283611723460971434588819353, −6.52924606105175789264379330467, −6.10729799087804802878345630821, −4.99384795132531131375581761845, −4.22695924763745898267207765160, −3.66521784509302450539583517295, −2.49514950227969214407742663473, −1.57122811404988768961709164080, 0, 1.57122811404988768961709164080, 2.49514950227969214407742663473, 3.66521784509302450539583517295, 4.22695924763745898267207765160, 4.99384795132531131375581761845, 6.10729799087804802878345630821, 6.52924606105175789264379330467, 7.53283611723460971434588819353, 8.533066462293558207418474625831

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