Properties

Label 2-3549-1.1-c1-0-58
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.54·2-s − 3-s + 4.49·4-s − 3.49·5-s + 2.54·6-s + 7-s − 6.35·8-s + 9-s + 8.90·10-s − 0.708·11-s − 4.49·12-s − 2.54·14-s + 3.49·15-s + 7.20·16-s − 7.09·17-s − 2.54·18-s + 0.311·19-s − 15.6·20-s − 21-s + 1.80·22-s + 7.88·23-s + 6.35·24-s + 7.20·25-s − 27-s + 4.49·28-s + 5.29·29-s − 8.90·30-s + ⋯
L(s)  = 1  − 1.80·2-s − 0.577·3-s + 2.24·4-s − 1.56·5-s + 1.04·6-s + 0.377·7-s − 2.24·8-s + 0.333·9-s + 2.81·10-s − 0.213·11-s − 1.29·12-s − 0.681·14-s + 0.901·15-s + 1.80·16-s − 1.72·17-s − 0.600·18-s + 0.0714·19-s − 3.50·20-s − 0.218·21-s + 0.384·22-s + 1.64·23-s + 1.29·24-s + 1.44·25-s − 0.192·27-s + 0.849·28-s + 0.983·29-s − 1.62·30-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: \(1\)
Selberg data: \((2,\ 3549,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.49T + 5T^{2} \)
11 \( 1 + 0.708T + 11T^{2} \)
17 \( 1 + 7.09T + 17T^{2} \)
19 \( 1 - 0.311T + 19T^{2} \)
23 \( 1 - 7.88T + 23T^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 + 7.29T + 31T^{2} \)
37 \( 1 - 1.41T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 3.29T + 43T^{2} \)
47 \( 1 - 6.11T + 47T^{2} \)
53 \( 1 + 3.72T + 53T^{2} \)
59 \( 1 + 2.19T + 59T^{2} \)
61 \( 1 - 2.51T + 61T^{2} \)
67 \( 1 - 9.17T + 67T^{2} \)
71 \( 1 + 0.708T + 71T^{2} \)
73 \( 1 - 5.21T + 73T^{2} \)
79 \( 1 + 2.78T + 79T^{2} \)
83 \( 1 - 6.11T + 83T^{2} \)
89 \( 1 - 11.1T + 89T^{2} \)
97 \( 1 - 7.79T + 97T^{2} \)
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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

−8.328241390276385854477290434768, −7.54312513864137711839901157372, −7.00903684918082879267724429859, −6.51028124538714843696680118775, −5.15726104353520628918801009581, −4.36743020913385218242698887899, −3.25870741364240731578751295095, −2.14024274886916768701661634990, −0.907735287282731237559980362312, 0, 0.907735287282731237559980362312, 2.14024274886916768701661634990, 3.25870741364240731578751295095, 4.36743020913385218242698887899, 5.15726104353520628918801009581, 6.51028124538714843696680118775, 7.00903684918082879267724429859, 7.54312513864137711839901157372, 8.328241390276385854477290434768

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