Properties

Label 2-3549-1.1-c1-0-83
Degree $2$
Conductor $3549$
Sign $-1$
Analytic cond. $28.3389$
Root an. cond. $5.32343$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s − 3·5-s − 2·6-s − 7-s + 9-s + 6·10-s + 2·12-s + 2·14-s − 3·15-s − 4·16-s + 2·17-s − 2·18-s − 19-s − 6·20-s − 21-s + 23-s + 4·25-s + 27-s − 2·28-s + 5·29-s + 6·30-s − 5·31-s + 8·32-s − 4·34-s + 3·35-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 1.34·5-s − 0.816·6-s − 0.377·7-s + 1/3·9-s + 1.89·10-s + 0.577·12-s + 0.534·14-s − 0.774·15-s − 16-s + 0.485·17-s − 0.471·18-s − 0.229·19-s − 1.34·20-s − 0.218·21-s + 0.208·23-s + 4/5·25-s + 0.192·27-s − 0.377·28-s + 0.928·29-s + 1.09·30-s − 0.898·31-s + 1.41·32-s − 0.685·34-s + 0.507·35-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: yes
Arithmetic: yes
Character: $\chi_{3549} (1, \cdot )$
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 \( 1 + p T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−8.291337964577720542000036601017, −7.59632976338319477439809420770, −7.23113173664801018219950626461, −6.36768099124137407662455495596, −5.04305933288398388366326087917, −4.09634718745809111617687686594, −3.40040765767289635727722931599, −2.35584928239184963657299205645, −1.11612014257101683402638604868, 0, 1.11612014257101683402638604868, 2.35584928239184963657299205645, 3.40040765767289635727722931599, 4.09634718745809111617687686594, 5.04305933288398388366326087917, 6.36768099124137407662455495596, 7.23113173664801018219950626461, 7.59632976338319477439809420770, 8.291337964577720542000036601017

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