Properties

Label 2-3549-1.1-c1-0-77
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  + 1.09·2-s + 3-s − 0.803·4-s + 3.36·5-s + 1.09·6-s − 7-s − 3.06·8-s + 9-s + 3.68·10-s + 0.408·11-s − 0.803·12-s − 1.09·14-s + 3.36·15-s − 1.74·16-s + 0.412·17-s + 1.09·18-s − 1.06·19-s − 2.70·20-s − 21-s + 0.446·22-s + 8.83·23-s − 3.06·24-s + 6.35·25-s + 27-s + 0.803·28-s − 3.54·29-s + 3.68·30-s + ⋯
L(s)  = 1  + 0.773·2-s + 0.577·3-s − 0.401·4-s + 1.50·5-s + 0.446·6-s − 0.377·7-s − 1.08·8-s + 0.333·9-s + 1.16·10-s + 0.123·11-s − 0.232·12-s − 0.292·14-s + 0.870·15-s − 0.436·16-s + 0.0999·17-s + 0.257·18-s − 0.244·19-s − 0.605·20-s − 0.218·21-s + 0.0951·22-s + 1.84·23-s − 0.625·24-s + 1.27·25-s + 0.192·27-s + 0.151·28-s − 0.658·29-s + 0.672·30-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

\(L(1)\) \(\approx\) \(3.856221742\)
\(L(\frac12)\) \(\approx\) \(3.856221742\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.09T + 2T^{2} \)
5 \( 1 - 3.36T + 5T^{2} \)
11 \( 1 - 0.408T + 11T^{2} \)
17 \( 1 - 0.412T + 17T^{2} \)
19 \( 1 + 1.06T + 19T^{2} \)
23 \( 1 - 8.83T + 23T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
31 \( 1 - 1.89T + 31T^{2} \)
37 \( 1 - 4.79T + 37T^{2} \)
41 \( 1 - 10.0T + 41T^{2} \)
43 \( 1 + 2.32T + 43T^{2} \)
47 \( 1 - 8.84T + 47T^{2} \)
53 \( 1 + 4.18T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 - 2.39T + 67T^{2} \)
71 \( 1 - 14.6T + 71T^{2} \)
73 \( 1 + 14.5T + 73T^{2} \)
79 \( 1 - 4.27T + 79T^{2} \)
83 \( 1 + 4.60T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−8.948189068102398529308354953631, −7.83911840830353451320262274392, −6.83204093721481962615364145060, −6.16508959806713784518759076672, −5.51088264818653255684098950611, −4.80869632535298161377533616893, −3.91877993871686463633205284175, −2.97019551383145003054425203839, −2.36175204935250091741360447612, −1.05749624545199054279569237558, 1.05749624545199054279569237558, 2.36175204935250091741360447612, 2.97019551383145003054425203839, 3.91877993871686463633205284175, 4.80869632535298161377533616893, 5.51088264818653255684098950611, 6.16508959806713784518759076672, 6.83204093721481962615364145060, 7.83911840830353451320262274392, 8.948189068102398529308354953631

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