Properties

Label 2-3549-1.1-c1-0-66
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  − 2.80·2-s + 3-s + 5.86·4-s + 3.14·5-s − 2.80·6-s + 7-s − 10.8·8-s + 9-s − 8.81·10-s − 3.00·11-s + 5.86·12-s − 2.80·14-s + 3.14·15-s + 18.6·16-s + 3.41·17-s − 2.80·18-s + 4.86·19-s + 18.4·20-s + 21-s + 8.42·22-s + 5.37·23-s − 10.8·24-s + 4.87·25-s + 27-s + 5.86·28-s + 0.848·29-s − 8.81·30-s + ⋯
L(s)  = 1  − 1.98·2-s + 0.577·3-s + 2.93·4-s + 1.40·5-s − 1.14·6-s + 0.377·7-s − 3.82·8-s + 0.333·9-s − 2.78·10-s − 0.905·11-s + 1.69·12-s − 0.749·14-s + 0.811·15-s + 4.65·16-s + 0.828·17-s − 0.660·18-s + 1.11·19-s + 4.11·20-s + 0.218·21-s + 1.79·22-s + 1.12·23-s − 2.20·24-s + 0.975·25-s + 0.192·27-s + 1.10·28-s + 0.157·29-s − 1.60·30-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

\(L(1)\) \(\approx\) \(1.436522459\)
\(L(\frac12)\) \(\approx\) \(1.436522459\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.80T + 2T^{2} \)
5 \( 1 - 3.14T + 5T^{2} \)
11 \( 1 + 3.00T + 11T^{2} \)
17 \( 1 - 3.41T + 17T^{2} \)
19 \( 1 - 4.86T + 19T^{2} \)
23 \( 1 - 5.37T + 23T^{2} \)
29 \( 1 - 0.848T + 29T^{2} \)
31 \( 1 + 0.451T + 31T^{2} \)
37 \( 1 - 3.88T + 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 - 7.20T + 43T^{2} \)
47 \( 1 - 9.33T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 1.45T + 59T^{2} \)
61 \( 1 - 4.95T + 61T^{2} \)
67 \( 1 + 4.57T + 67T^{2} \)
71 \( 1 - 2.36T + 71T^{2} \)
73 \( 1 + 1.24T + 73T^{2} \)
79 \( 1 + 7.10T + 79T^{2} \)
83 \( 1 + 5.04T + 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 - 4.29T + 97T^{2} \)
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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

−8.782418594053004208438597737709, −7.85643825889488090475841732615, −7.49938625580337616227275641139, −6.64888286981612697283821718294, −5.79213289026856160569104802843, −5.17302199929246603699331804573, −3.19640912909862537369603689350, −2.61605628754538014217936482474, −1.75221411916394889174160106202, −0.975155668312607978050284230712, 0.975155668312607978050284230712, 1.75221411916394889174160106202, 2.61605628754538014217936482474, 3.19640912909862537369603689350, 5.17302199929246603699331804573, 5.79213289026856160569104802843, 6.64888286981612697283821718294, 7.49938625580337616227275641139, 7.85643825889488090475841732615, 8.782418594053004208438597737709

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