Properties

Label 2-3549-1.1-c1-0-105
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.311·2-s + 3-s − 1.90·4-s − 1.52·5-s + 0.311·6-s − 7-s − 1.21·8-s + 9-s − 0.474·10-s − 1.09·11-s − 1.90·12-s − 0.311·14-s − 1.52·15-s + 3.42·16-s + 4.42·17-s + 0.311·18-s + 1.80·19-s + 2.90·20-s − 21-s − 0.341·22-s + 3.80·23-s − 1.21·24-s − 2.67·25-s + 27-s + 1.90·28-s − 0.755·29-s − 0.474·30-s + ⋯
L(s)  = 1  + 0.219·2-s + 0.577·3-s − 0.951·4-s − 0.682·5-s + 0.127·6-s − 0.377·7-s − 0.429·8-s + 0.333·9-s − 0.150·10-s − 0.330·11-s − 0.549·12-s − 0.0831·14-s − 0.393·15-s + 0.857·16-s + 1.07·17-s + 0.0733·18-s + 0.414·19-s + 0.649·20-s − 0.218·21-s − 0.0727·22-s + 0.793·23-s − 0.247·24-s − 0.534·25-s + 0.192·27-s + 0.359·28-s − 0.140·29-s − 0.0866·30-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 \( 1 - 0.311T + 2T^{2} \)
5 \( 1 + 1.52T + 5T^{2} \)
11 \( 1 + 1.09T + 11T^{2} \)
17 \( 1 - 4.42T + 17T^{2} \)
19 \( 1 - 1.80T + 19T^{2} \)
23 \( 1 - 3.80T + 23T^{2} \)
29 \( 1 + 0.755T + 29T^{2} \)
31 \( 1 - 4.85T + 31T^{2} \)
37 \( 1 + 5.80T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 - 5.24T + 43T^{2} \)
47 \( 1 + 2.28T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 0.474T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 + 9.80T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 - 3.47T + 73T^{2} \)
79 \( 1 + 5.37T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + 4.42T + 97T^{2} \)
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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

−8.191704082940168614541530754681, −7.66969512564356550043956271443, −6.83880109159493490394988219385, −5.76848489580503847188756132254, −5.04330297313452013854579138019, −4.25290270243069877042342930923, −3.41927488060824146090690480056, −2.95977847728260833218594699578, −1.34517993681903708418920664535, 0, 1.34517993681903708418920664535, 2.95977847728260833218594699578, 3.41927488060824146090690480056, 4.25290270243069877042342930923, 5.04330297313452013854579138019, 5.76848489580503847188756132254, 6.83880109159493490394988219385, 7.66969512564356550043956271443, 8.191704082940168614541530754681

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