Properties

Label 2-3549-1.1-c1-0-34
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  + 1.90·2-s − 3-s + 1.61·4-s − 1.76·5-s − 1.90·6-s − 7-s − 0.738·8-s + 9-s − 3.35·10-s + 3.13·11-s − 1.61·12-s − 1.90·14-s + 1.76·15-s − 4.62·16-s − 0.934·17-s + 1.90·18-s − 2.13·19-s − 2.84·20-s + 21-s + 5.95·22-s + 5.87·23-s + 0.738·24-s − 1.88·25-s − 27-s − 1.61·28-s − 2.37·29-s + 3.35·30-s + ⋯
L(s)  = 1  + 1.34·2-s − 0.577·3-s + 0.805·4-s − 0.789·5-s − 0.775·6-s − 0.377·7-s − 0.260·8-s + 0.333·9-s − 1.06·10-s + 0.944·11-s − 0.465·12-s − 0.507·14-s + 0.455·15-s − 1.15·16-s − 0.226·17-s + 0.447·18-s − 0.489·19-s − 0.636·20-s + 0.218·21-s + 1.26·22-s + 1.22·23-s + 0.150·24-s − 0.376·25-s − 0.192·27-s − 0.304·28-s − 0.440·29-s + 0.612·30-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

\(L(1)\) \(\approx\) \(2.297836016\)
\(L(\frac12)\) \(\approx\) \(2.297836016\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.90T + 2T^{2} \)
5 \( 1 + 1.76T + 5T^{2} \)
11 \( 1 - 3.13T + 11T^{2} \)
17 \( 1 + 0.934T + 17T^{2} \)
19 \( 1 + 2.13T + 19T^{2} \)
23 \( 1 - 5.87T + 23T^{2} \)
29 \( 1 + 2.37T + 29T^{2} \)
31 \( 1 - 6.06T + 31T^{2} \)
37 \( 1 - 2.02T + 37T^{2} \)
41 \( 1 - 6.16T + 41T^{2} \)
43 \( 1 - 5.68T + 43T^{2} \)
47 \( 1 + 5.26T + 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 4.83T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 - 6.65T + 67T^{2} \)
71 \( 1 - 7.17T + 71T^{2} \)
73 \( 1 + 4.85T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 + 8.30T + 89T^{2} \)
97 \( 1 - 18.2T + 97T^{2} \)
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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

−8.588919033662996136353150001496, −7.51473479687875336761470727552, −6.76487371606439775244328250884, −6.24450797339499275199498067324, −5.47280667367365075646840893585, −4.59729922770899839433096630056, −4.06002706983314882545904230989, −3.40435432547816981891626334372, −2.35226775679900831600469597089, −0.74123019530914175773919584634, 0.74123019530914175773919584634, 2.35226775679900831600469597089, 3.40435432547816981891626334372, 4.06002706983314882545904230989, 4.59729922770899839433096630056, 5.47280667367365075646840893585, 6.24450797339499275199498067324, 6.76487371606439775244328250884, 7.51473479687875336761470727552, 8.588919033662996136353150001496

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