Properties

Label 2-3549-1.1-c1-0-132
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  + 2.49·2-s + 3-s + 4.24·4-s + 2.35·5-s + 2.49·6-s − 7-s + 5.59·8-s + 9-s + 5.87·10-s + 5.16·11-s + 4.24·12-s − 2.49·14-s + 2.35·15-s + 5.50·16-s − 1.70·17-s + 2.49·18-s − 8.02·19-s + 9.97·20-s − 21-s + 12.9·22-s + 4.21·23-s + 5.59·24-s + 0.531·25-s + 27-s − 4.24·28-s − 5.24·29-s + 5.87·30-s + ⋯
L(s)  = 1  + 1.76·2-s + 0.577·3-s + 2.12·4-s + 1.05·5-s + 1.01·6-s − 0.377·7-s + 1.97·8-s + 0.333·9-s + 1.85·10-s + 1.55·11-s + 1.22·12-s − 0.667·14-s + 0.607·15-s + 1.37·16-s − 0.413·17-s + 0.588·18-s − 1.84·19-s + 2.23·20-s − 0.218·21-s + 2.75·22-s + 0.879·23-s + 1.14·24-s + 0.106·25-s + 0.192·27-s − 0.801·28-s − 0.973·29-s + 1.07·30-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

\(L(1)\) \(\approx\) \(8.467003606\)
\(L(\frac12)\) \(\approx\) \(8.467003606\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.49T + 2T^{2} \)
5 \( 1 - 2.35T + 5T^{2} \)
11 \( 1 - 5.16T + 11T^{2} \)
17 \( 1 + 1.70T + 17T^{2} \)
19 \( 1 + 8.02T + 19T^{2} \)
23 \( 1 - 4.21T + 23T^{2} \)
29 \( 1 + 5.24T + 29T^{2} \)
31 \( 1 - 1.11T + 31T^{2} \)
37 \( 1 - 8.21T + 37T^{2} \)
41 \( 1 + 6.75T + 41T^{2} \)
43 \( 1 + 0.965T + 43T^{2} \)
47 \( 1 - 9.18T + 47T^{2} \)
53 \( 1 - 11.9T + 53T^{2} \)
59 \( 1 + 2.98T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 - 0.318T + 71T^{2} \)
73 \( 1 + 0.542T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 - 5.79T + 83T^{2} \)
89 \( 1 + 7.80T + 89T^{2} \)
97 \( 1 + 0.287T + 97T^{2} \)
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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

−8.803891482133730597591956097746, −7.41640384026178328257071100893, −6.69254558251519150187045456019, −6.20514981282258158202313990130, −5.64516200282490258602364092185, −4.45286480802641267558809038418, −4.10475204480997034608749576083, −3.14162080721660027329249602775, −2.30049224857753068982409524250, −1.58772784844415169439332878941, 1.58772784844415169439332878941, 2.30049224857753068982409524250, 3.14162080721660027329249602775, 4.10475204480997034608749576083, 4.45286480802641267558809038418, 5.64516200282490258602364092185, 6.20514981282258158202313990130, 6.69254558251519150187045456019, 7.41640384026178328257071100893, 8.803891482133730597591956097746

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