Properties

Label 2-3549-1.1-c1-0-140
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  + 1.65·2-s − 3-s + 0.726·4-s + 2.92·5-s − 1.65·6-s − 7-s − 2.10·8-s + 9-s + 4.82·10-s − 4.37·11-s − 0.726·12-s − 1.65·14-s − 2.92·15-s − 4.92·16-s + 3.65·17-s + 1.65·18-s − 5.75·19-s + 2.12·20-s + 21-s − 7.22·22-s + 7.02·23-s + 2.10·24-s + 3.55·25-s − 27-s − 0.726·28-s + 1.19·29-s − 4.82·30-s + ⋯
L(s)  = 1  + 1.16·2-s − 0.577·3-s + 0.363·4-s + 1.30·5-s − 0.674·6-s − 0.377·7-s − 0.743·8-s + 0.333·9-s + 1.52·10-s − 1.31·11-s − 0.209·12-s − 0.441·14-s − 0.755·15-s − 1.23·16-s + 0.885·17-s + 0.389·18-s − 1.32·19-s + 0.474·20-s + 0.218·21-s − 1.54·22-s + 1.46·23-s + 0.429·24-s + 0.711·25-s − 0.192·27-s − 0.137·28-s + 0.222·29-s − 0.881·30-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: \(1\)
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 \( 1 - 1.65T + 2T^{2} \)
5 \( 1 - 2.92T + 5T^{2} \)
11 \( 1 + 4.37T + 11T^{2} \)
17 \( 1 - 3.65T + 17T^{2} \)
19 \( 1 + 5.75T + 19T^{2} \)
23 \( 1 - 7.02T + 23T^{2} \)
29 \( 1 - 1.19T + 29T^{2} \)
31 \( 1 + 6.47T + 31T^{2} \)
37 \( 1 + 2.92T + 37T^{2} \)
41 \( 1 + 8.60T + 41T^{2} \)
43 \( 1 + 5.72T + 43T^{2} \)
47 \( 1 + 9.58T + 47T^{2} \)
53 \( 1 + 0.302T + 53T^{2} \)
59 \( 1 - 3.02T + 59T^{2} \)
61 \( 1 + 0.302T + 61T^{2} \)
67 \( 1 + 8.70T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 0.932T + 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 + 1.26T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−8.222670772099456133334353842692, −6.98749777033367017660818717495, −6.47939429052555343443938303947, −5.60334749735965075560252124079, −5.32233869505146113694213276181, −4.65008980045169729425752255918, −3.46678786206920536405287757848, −2.73782827203641924204809649823, −1.73049021298553151312014415776, 0, 1.73049021298553151312014415776, 2.73782827203641924204809649823, 3.46678786206920536405287757848, 4.65008980045169729425752255918, 5.32233869505146113694213276181, 5.60334749735965075560252124079, 6.47939429052555343443938303947, 6.98749777033367017660818717495, 8.222670772099456133334353842692

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