Properties

Label 2-3549-1.1-c1-0-151
Degree $2$
Conductor $3549$
Sign $-1$
Analytic cond. $28.3389$
Root an. cond. $5.32343$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 4-s + 2·5-s + 6-s + 7-s − 3·8-s + 9-s + 2·10-s − 4·11-s − 12-s + 14-s + 2·15-s − 16-s − 6·17-s + 18-s − 4·19-s − 2·20-s + 21-s − 4·22-s − 3·24-s − 25-s + 27-s − 28-s − 2·29-s + 2·30-s + 5·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.288·12-s + 0.267·14-s + 0.516·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s − 0.447·20-s + 0.218·21-s − 0.852·22-s − 0.612·24-s − 1/5·25-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.365·30-s + 0.883·32-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: yes
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 \( 1 - T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.407377139641223631410053626523, −7.48728380165555641417746344077, −6.50644747922743051012510731425, −5.81934189952643673303329908812, −5.00585213491191130205663070545, −4.48166630149180553918760034444, −3.51954643368349943520882598505, −2.53133608948261501639635653970, −1.88453572273281662045561783260, 0, 1.88453572273281662045561783260, 2.53133608948261501639635653970, 3.51954643368349943520882598505, 4.48166630149180553918760034444, 5.00585213491191130205663070545, 5.81934189952643673303329908812, 6.50644747922743051012510731425, 7.48728380165555641417746344077, 8.407377139641223631410053626523

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