Properties

Label 2-3549-1.1-c1-0-2
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  − 2.41·2-s + 3-s + 3.84·4-s − 1.98·5-s − 2.41·6-s − 7-s − 4.45·8-s + 9-s + 4.79·10-s − 5.56·11-s + 3.84·12-s + 2.41·14-s − 1.98·15-s + 3.07·16-s − 5.66·17-s − 2.41·18-s − 2.77·19-s − 7.61·20-s − 21-s + 13.4·22-s − 2.20·23-s − 4.45·24-s − 1.06·25-s + 27-s − 3.84·28-s − 5.79·29-s + 4.79·30-s + ⋯
L(s)  = 1  − 1.70·2-s + 0.577·3-s + 1.92·4-s − 0.886·5-s − 0.986·6-s − 0.377·7-s − 1.57·8-s + 0.333·9-s + 1.51·10-s − 1.67·11-s + 1.10·12-s + 0.645·14-s − 0.511·15-s + 0.768·16-s − 1.37·17-s − 0.569·18-s − 0.636·19-s − 1.70·20-s − 0.218·21-s + 2.86·22-s − 0.458·23-s − 0.908·24-s − 0.213·25-s + 0.192·27-s − 0.726·28-s − 1.07·29-s + 0.874·30-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

\(L(1)\) \(\approx\) \(0.2453735642\)
\(L(\frac12)\) \(\approx\) \(0.2453735642\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41T + 2T^{2} \)
5 \( 1 + 1.98T + 5T^{2} \)
11 \( 1 + 5.56T + 11T^{2} \)
17 \( 1 + 5.66T + 17T^{2} \)
19 \( 1 + 2.77T + 19T^{2} \)
23 \( 1 + 2.20T + 23T^{2} \)
29 \( 1 + 5.79T + 29T^{2} \)
31 \( 1 + 0.187T + 31T^{2} \)
37 \( 1 - 5.00T + 37T^{2} \)
41 \( 1 + 8.39T + 41T^{2} \)
43 \( 1 + 6.93T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 2.78T + 53T^{2} \)
59 \( 1 - 6.18T + 59T^{2} \)
61 \( 1 - 13.1T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 - 3.12T + 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + 6.24T + 89T^{2} \)
97 \( 1 - 0.650T + 97T^{2} \)
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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

−8.474793313593424769440906494325, −8.074644063095379925599153704824, −7.36512543410732956349220521030, −6.87969499210271111521913253120, −5.83544459521431687077728656670, −4.62724415864580241854557887015, −3.67164738693881587563700878491, −2.57543990869740904906702035717, −1.99926795703766382000004245444, −0.33999575021588458312404706749, 0.33999575021588458312404706749, 1.99926795703766382000004245444, 2.57543990869740904906702035717, 3.67164738693881587563700878491, 4.62724415864580241854557887015, 5.83544459521431687077728656670, 6.87969499210271111521913253120, 7.36512543410732956349220521030, 8.074644063095379925599153704824, 8.474793313593424769440906494325

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