Properties

Label 2-3549-1.1-c1-0-46
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  − 1.29·2-s + 3-s − 0.310·4-s + 1.99·5-s − 1.29·6-s − 7-s + 3.00·8-s + 9-s − 2.59·10-s + 2.23·11-s − 0.310·12-s + 1.29·14-s + 1.99·15-s − 3.28·16-s − 0.168·17-s − 1.29·18-s − 6.89·19-s − 0.618·20-s − 21-s − 2.90·22-s + 1.53·23-s + 3.00·24-s − 1.02·25-s + 27-s + 0.310·28-s + 8.81·29-s − 2.59·30-s + ⋯
L(s)  = 1  − 0.919·2-s + 0.577·3-s − 0.155·4-s + 0.891·5-s − 0.530·6-s − 0.377·7-s + 1.06·8-s + 0.333·9-s − 0.819·10-s + 0.673·11-s − 0.0896·12-s + 0.347·14-s + 0.514·15-s − 0.820·16-s − 0.0408·17-s − 0.306·18-s − 1.58·19-s − 0.138·20-s − 0.218·21-s − 0.619·22-s + 0.321·23-s + 0.613·24-s − 0.205·25-s + 0.192·27-s + 0.0586·28-s + 1.63·29-s − 0.473·30-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

\(L(1)\) \(\approx\) \(1.504056462\)
\(L(\frac12)\) \(\approx\) \(1.504056462\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.29T + 2T^{2} \)
5 \( 1 - 1.99T + 5T^{2} \)
11 \( 1 - 2.23T + 11T^{2} \)
17 \( 1 + 0.168T + 17T^{2} \)
19 \( 1 + 6.89T + 19T^{2} \)
23 \( 1 - 1.53T + 23T^{2} \)
29 \( 1 - 8.81T + 29T^{2} \)
31 \( 1 + 0.0996T + 31T^{2} \)
37 \( 1 - 4.91T + 37T^{2} \)
41 \( 1 - 0.944T + 41T^{2} \)
43 \( 1 - 7.75T + 43T^{2} \)
47 \( 1 + 1.08T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 - 9.17T + 59T^{2} \)
61 \( 1 + 3.27T + 61T^{2} \)
67 \( 1 + 8.51T + 67T^{2} \)
71 \( 1 - 6.19T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 0.338T + 83T^{2} \)
89 \( 1 + 7.96T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−8.730307556621612662437258021698, −8.125713372960219570682813794123, −7.17659094380694471510030819515, −6.50523960467027870297470715285, −5.73864488130272105787539428921, −4.55191902515733217640223001910, −4.00317440546214556870386854779, −2.70064107787577422091488595003, −1.89591726546386100534520123236, −0.834278871010852081893995281506, 0.834278871010852081893995281506, 1.89591726546386100534520123236, 2.70064107787577422091488595003, 4.00317440546214556870386854779, 4.55191902515733217640223001910, 5.73864488130272105787539428921, 6.50523960467027870297470715285, 7.17659094380694471510030819515, 8.125713372960219570682813794123, 8.730307556621612662437258021698

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