Properties

Label 2-3549-1.1-c1-0-17
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  + 0.146·2-s − 3-s − 1.97·4-s + 1.04·5-s − 0.146·6-s − 7-s − 0.581·8-s + 9-s + 0.152·10-s − 3.53·11-s + 1.97·12-s − 0.146·14-s − 1.04·15-s + 3.87·16-s + 6.85·17-s + 0.146·18-s + 3.55·19-s − 2.06·20-s + 21-s − 0.516·22-s + 0.0937·23-s + 0.581·24-s − 3.91·25-s − 27-s + 1.97·28-s − 10.5·29-s − 0.152·30-s + ⋯
L(s)  = 1  + 0.103·2-s − 0.577·3-s − 0.989·4-s + 0.466·5-s − 0.0596·6-s − 0.377·7-s − 0.205·8-s + 0.333·9-s + 0.0482·10-s − 1.06·11-s + 0.571·12-s − 0.0390·14-s − 0.269·15-s + 0.968·16-s + 1.66·17-s + 0.0344·18-s + 0.814·19-s − 0.461·20-s + 0.218·21-s − 0.110·22-s + 0.0195·23-s + 0.118·24-s − 0.782·25-s − 0.192·27-s + 0.373·28-s − 1.96·29-s − 0.0278·30-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

\(L(1)\) \(\approx\) \(0.9572521572\)
\(L(\frac12)\) \(\approx\) \(0.9572521572\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.146T + 2T^{2} \)
5 \( 1 - 1.04T + 5T^{2} \)
11 \( 1 + 3.53T + 11T^{2} \)
17 \( 1 - 6.85T + 17T^{2} \)
19 \( 1 - 3.55T + 19T^{2} \)
23 \( 1 - 0.0937T + 23T^{2} \)
29 \( 1 + 10.5T + 29T^{2} \)
31 \( 1 + 4.44T + 31T^{2} \)
37 \( 1 + 12.0T + 37T^{2} \)
41 \( 1 + 1.37T + 41T^{2} \)
43 \( 1 - 6.68T + 43T^{2} \)
47 \( 1 - 9.55T + 47T^{2} \)
53 \( 1 + 3.01T + 53T^{2} \)
59 \( 1 + 0.747T + 59T^{2} \)
61 \( 1 - 5.72T + 61T^{2} \)
67 \( 1 - 9.38T + 67T^{2} \)
71 \( 1 + 4.41T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 0.520T + 83T^{2} \)
89 \( 1 - 4.78T + 89T^{2} \)
97 \( 1 + 5.89T + 97T^{2} \)
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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

−8.661501795411068949268154921949, −7.62922763841549387507879352442, −7.32431303684221376086393565619, −5.87758025674721062139913443194, −5.55056839639382088994214401441, −5.07154200841789528057426898723, −3.83788527711345913435509997033, −3.27804776965463905262742928138, −1.88658718603194916639427147836, −0.58402048596228891002918675683, 0.58402048596228891002918675683, 1.88658718603194916639427147836, 3.27804776965463905262742928138, 3.83788527711345913435509997033, 5.07154200841789528057426898723, 5.55056839639382088994214401441, 5.87758025674721062139913443194, 7.32431303684221376086393565619, 7.62922763841549387507879352442, 8.661501795411068949268154921949

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