Properties

Label 2-3549-1.1-c1-0-125
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  + 2.07·2-s + 3-s + 2.29·4-s + 3.98·5-s + 2.07·6-s − 7-s + 0.606·8-s + 9-s + 8.25·10-s + 2.61·11-s + 2.29·12-s − 2.07·14-s + 3.98·15-s − 3.32·16-s + 2.46·17-s + 2.07·18-s + 5.30·19-s + 9.13·20-s − 21-s + 5.41·22-s − 8.91·23-s + 0.606·24-s + 10.8·25-s + 27-s − 2.29·28-s − 3.55·29-s + 8.25·30-s + ⋯
L(s)  = 1  + 1.46·2-s + 0.577·3-s + 1.14·4-s + 1.78·5-s + 0.845·6-s − 0.377·7-s + 0.214·8-s + 0.333·9-s + 2.61·10-s + 0.788·11-s + 0.661·12-s − 0.553·14-s + 1.02·15-s − 0.832·16-s + 0.597·17-s + 0.488·18-s + 1.21·19-s + 2.04·20-s − 0.218·21-s + 1.15·22-s − 1.85·23-s + 0.123·24-s + 2.17·25-s + 0.192·27-s − 0.433·28-s − 0.659·29-s + 1.50·30-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

\(L(1)\) \(\approx\) \(7.162972355\)
\(L(\frac12)\) \(\approx\) \(7.162972355\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.07T + 2T^{2} \)
5 \( 1 - 3.98T + 5T^{2} \)
11 \( 1 - 2.61T + 11T^{2} \)
17 \( 1 - 2.46T + 17T^{2} \)
19 \( 1 - 5.30T + 19T^{2} \)
23 \( 1 + 8.91T + 23T^{2} \)
29 \( 1 + 3.55T + 29T^{2} \)
31 \( 1 - 6.25T + 31T^{2} \)
37 \( 1 + 3.72T + 37T^{2} \)
41 \( 1 + 6.88T + 41T^{2} \)
43 \( 1 - 7.59T + 43T^{2} \)
47 \( 1 + 2.43T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 0.944T + 59T^{2} \)
61 \( 1 - 4.95T + 61T^{2} \)
67 \( 1 + 16.0T + 67T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 - 3.16T + 73T^{2} \)
79 \( 1 + 5.15T + 79T^{2} \)
83 \( 1 - 1.47T + 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 + 4.52T + 97T^{2} \)
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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

−8.701138045606361951668046217481, −7.63020781285858188681736647502, −6.62722358142648720399654732706, −6.16818365097082517337463476732, −5.55607364016173256535590050736, −4.83814785408230817294901949562, −3.82261979382027885780957155579, −3.14115707654893824107783820018, −2.29408377158861586124821213803, −1.46670342261174121861042560300, 1.46670342261174121861042560300, 2.29408377158861586124821213803, 3.14115707654893824107783820018, 3.82261979382027885780957155579, 4.83814785408230817294901949562, 5.55607364016173256535590050736, 6.16818365097082517337463476732, 6.62722358142648720399654732706, 7.63020781285858188681736647502, 8.701138045606361951668046217481

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