Properties

Label 2-3549-1.1-c1-0-50
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  + 2.54·2-s − 3-s + 4.46·4-s − 4.07·5-s − 2.54·6-s − 7-s + 6.26·8-s + 9-s − 10.3·10-s + 3.87·11-s − 4.46·12-s − 2.54·14-s + 4.07·15-s + 6.99·16-s − 2.47·17-s + 2.54·18-s + 2.67·19-s − 18.1·20-s + 21-s + 9.85·22-s − 4.84·23-s − 6.26·24-s + 11.5·25-s − 27-s − 4.46·28-s + 5.76·29-s + 10.3·30-s + ⋯
L(s)  = 1  + 1.79·2-s − 0.577·3-s + 2.23·4-s − 1.82·5-s − 1.03·6-s − 0.377·7-s + 2.21·8-s + 0.333·9-s − 3.27·10-s + 1.16·11-s − 1.28·12-s − 0.679·14-s + 1.05·15-s + 1.74·16-s − 0.599·17-s + 0.599·18-s + 0.614·19-s − 4.06·20-s + 0.218·21-s + 2.10·22-s − 1.01·23-s − 1.27·24-s + 2.31·25-s − 0.192·27-s − 0.843·28-s + 1.07·29-s + 1.89·30-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

\(L(1)\) \(\approx\) \(3.470578185\)
\(L(\frac12)\) \(\approx\) \(3.470578185\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.54T + 2T^{2} \)
5 \( 1 + 4.07T + 5T^{2} \)
11 \( 1 - 3.87T + 11T^{2} \)
17 \( 1 + 2.47T + 17T^{2} \)
19 \( 1 - 2.67T + 19T^{2} \)
23 \( 1 + 4.84T + 23T^{2} \)
29 \( 1 - 5.76T + 29T^{2} \)
31 \( 1 - 4.35T + 31T^{2} \)
37 \( 1 + 2.25T + 37T^{2} \)
41 \( 1 - 4.84T + 41T^{2} \)
43 \( 1 - 3.69T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 8.39T + 59T^{2} \)
61 \( 1 + 7.02T + 61T^{2} \)
67 \( 1 + 5.13T + 67T^{2} \)
71 \( 1 - 9.80T + 71T^{2} \)
73 \( 1 - 8.54T + 73T^{2} \)
79 \( 1 - 0.0251T + 79T^{2} \)
83 \( 1 - 0.202T + 83T^{2} \)
89 \( 1 - 18.2T + 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−8.273792906723466976179659238974, −7.42611586676430139108548598332, −6.82781169279905947031736151440, −6.30030730803225577602639208654, −5.40113549539353332844232611346, −4.44120955544396818367495253504, −4.09428511531809086272320046392, −3.49866147104178434496844779615, −2.49759674741467857607450609790, −0.868216343809721996099709666829, 0.868216343809721996099709666829, 2.49759674741467857607450609790, 3.49866147104178434496844779615, 4.09428511531809086272320046392, 4.44120955544396818367495253504, 5.40113549539353332844232611346, 6.30030730803225577602639208654, 6.82781169279905947031736151440, 7.42611586676430139108548598332, 8.273792906723466976179659238974

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