Properties

Label 2-3549-1.1-c1-0-7
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  − 1.36·2-s − 3-s − 0.138·4-s − 3.32·5-s + 1.36·6-s − 7-s + 2.91·8-s + 9-s + 4.53·10-s + 1.11·11-s + 0.138·12-s + 1.36·14-s + 3.32·15-s − 3.70·16-s + 0.665·17-s − 1.36·18-s + 3.22·19-s + 0.458·20-s + 21-s − 1.51·22-s + 5.07·23-s − 2.91·24-s + 6.02·25-s − 27-s + 0.138·28-s − 0.615·29-s − 4.53·30-s + ⋯
L(s)  = 1  − 0.964·2-s − 0.577·3-s − 0.0690·4-s − 1.48·5-s + 0.557·6-s − 0.377·7-s + 1.03·8-s + 0.333·9-s + 1.43·10-s + 0.335·11-s + 0.0398·12-s + 0.364·14-s + 0.857·15-s − 0.926·16-s + 0.161·17-s − 0.321·18-s + 0.740·19-s + 0.102·20-s + 0.218·21-s − 0.324·22-s + 1.05·23-s − 0.595·24-s + 1.20·25-s − 0.192·27-s + 0.0261·28-s − 0.114·29-s − 0.827·30-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

\(L(1)\) \(\approx\) \(0.3518583434\)
\(L(\frac12)\) \(\approx\) \(0.3518583434\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.36T + 2T^{2} \)
5 \( 1 + 3.32T + 5T^{2} \)
11 \( 1 - 1.11T + 11T^{2} \)
17 \( 1 - 0.665T + 17T^{2} \)
19 \( 1 - 3.22T + 19T^{2} \)
23 \( 1 - 5.07T + 23T^{2} \)
29 \( 1 + 0.615T + 29T^{2} \)
31 \( 1 + 8.84T + 31T^{2} \)
37 \( 1 + 3.79T + 37T^{2} \)
41 \( 1 - 0.916T + 41T^{2} \)
43 \( 1 - 8.09T + 43T^{2} \)
47 \( 1 + 13.0T + 47T^{2} \)
53 \( 1 - 3.52T + 53T^{2} \)
59 \( 1 - 5.48T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 + 3.47T + 67T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 + 7.86T + 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + 1.42T + 97T^{2} \)
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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

−8.736216496775405276139616256421, −7.58422074798024713172416194573, −7.49721589321144736844836087903, −6.64801736183355672276067483389, −5.49460725932256146242477040811, −4.69432529264963244911527930731, −3.93231717312042436385909792160, −3.17294307055690495076610215020, −1.48337139677062981704276590571, −0.44619932091872179528723609212, 0.44619932091872179528723609212, 1.48337139677062981704276590571, 3.17294307055690495076610215020, 3.93231717312042436385909792160, 4.69432529264963244911527930731, 5.49460725932256146242477040811, 6.64801736183355672276067483389, 7.49721589321144736844836087903, 7.58422074798024713172416194573, 8.736216496775405276139616256421

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