Properties

Label 2-3549-1.1-c1-0-142
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  + 2.61·2-s + 3-s + 4.83·4-s + 4.09·5-s + 2.61·6-s + 7-s + 7.42·8-s + 9-s + 10.7·10-s − 5.70·11-s + 4.83·12-s + 2.61·14-s + 4.09·15-s + 9.73·16-s − 4.37·17-s + 2.61·18-s − 3.61·19-s + 19.8·20-s + 21-s − 14.9·22-s + 1.52·23-s + 7.42·24-s + 11.7·25-s + 27-s + 4.83·28-s − 8.65·29-s + 10.7·30-s + ⋯
L(s)  = 1  + 1.84·2-s + 0.577·3-s + 2.41·4-s + 1.83·5-s + 1.06·6-s + 0.377·7-s + 2.62·8-s + 0.333·9-s + 3.38·10-s − 1.71·11-s + 1.39·12-s + 0.698·14-s + 1.05·15-s + 2.43·16-s − 1.06·17-s + 0.616·18-s − 0.828·19-s + 4.43·20-s + 0.218·21-s − 3.17·22-s + 0.318·23-s + 1.51·24-s + 2.35·25-s + 0.192·27-s + 0.914·28-s − 1.60·29-s + 1.95·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.714402942\)
\(L(\frac12)\) \(\approx\) \(9.714402942\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.61T + 2T^{2} \)
5 \( 1 - 4.09T + 5T^{2} \)
11 \( 1 + 5.70T + 11T^{2} \)
17 \( 1 + 4.37T + 17T^{2} \)
19 \( 1 + 3.61T + 19T^{2} \)
23 \( 1 - 1.52T + 23T^{2} \)
29 \( 1 + 8.65T + 29T^{2} \)
31 \( 1 - 2.53T + 31T^{2} \)
37 \( 1 + 0.399T + 37T^{2} \)
41 \( 1 + 3.49T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + 2.38T + 47T^{2} \)
53 \( 1 + 4.72T + 53T^{2} \)
59 \( 1 - 3.98T + 59T^{2} \)
61 \( 1 + 1.11T + 61T^{2} \)
67 \( 1 - 8.84T + 67T^{2} \)
71 \( 1 - 4.15T + 71T^{2} \)
73 \( 1 - 6.08T + 73T^{2} \)
79 \( 1 + 0.225T + 79T^{2} \)
83 \( 1 + 9.32T + 83T^{2} \)
89 \( 1 + 9.60T + 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−8.475666150596723238755990265471, −7.54386318546340417455314574996, −6.75262449953668851585836437565, −6.07765470596631906272307592423, −5.35676337256959815667834336452, −4.94951802118450841201315812431, −4.03588123846257837737450708634, −2.83897026760403742587017892312, −2.34760189318594660204717216627, −1.78736745059021445201417036359, 1.78736745059021445201417036359, 2.34760189318594660204717216627, 2.83897026760403742587017892312, 4.03588123846257837737450708634, 4.94951802118450841201315812431, 5.35676337256959815667834336452, 6.07765470596631906272307592423, 6.75262449953668851585836437565, 7.54386318546340417455314574996, 8.475666150596723238755990265471

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