Properties

Label 2-3549-1.1-c1-0-37
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  − 1.08·2-s + 3-s − 0.830·4-s − 1.66·5-s − 1.08·6-s − 7-s + 3.06·8-s + 9-s + 1.80·10-s + 4.63·11-s − 0.830·12-s + 1.08·14-s − 1.66·15-s − 1.64·16-s + 4.03·17-s − 1.08·18-s + 0.793·19-s + 1.38·20-s − 21-s − 5.01·22-s + 4.31·23-s + 3.06·24-s − 2.21·25-s + 27-s + 0.830·28-s + 4.61·29-s + 1.80·30-s + ⋯
L(s)  = 1  − 0.764·2-s + 0.577·3-s − 0.415·4-s − 0.746·5-s − 0.441·6-s − 0.377·7-s + 1.08·8-s + 0.333·9-s + 0.570·10-s + 1.39·11-s − 0.239·12-s + 0.288·14-s − 0.430·15-s − 0.411·16-s + 0.977·17-s − 0.254·18-s + 0.182·19-s + 0.310·20-s − 0.218·21-s − 1.06·22-s + 0.899·23-s + 0.624·24-s − 0.442·25-s + 0.192·27-s + 0.157·28-s + 0.857·29-s + 0.329·30-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

\(L(1)\) \(\approx\) \(1.201686757\)
\(L(\frac12)\) \(\approx\) \(1.201686757\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.08T + 2T^{2} \)
5 \( 1 + 1.66T + 5T^{2} \)
11 \( 1 - 4.63T + 11T^{2} \)
17 \( 1 - 4.03T + 17T^{2} \)
19 \( 1 - 0.793T + 19T^{2} \)
23 \( 1 - 4.31T + 23T^{2} \)
29 \( 1 - 4.61T + 29T^{2} \)
31 \( 1 + 5.75T + 31T^{2} \)
37 \( 1 + 5.73T + 37T^{2} \)
41 \( 1 + 7.33T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 5.10T + 47T^{2} \)
53 \( 1 + 6.41T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 1.86T + 67T^{2} \)
71 \( 1 + 6.08T + 71T^{2} \)
73 \( 1 - 3.11T + 73T^{2} \)
79 \( 1 - 2.28T + 79T^{2} \)
83 \( 1 + 0.655T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−8.705840044240023545250390625473, −7.912467067480556210729300376728, −7.34419981263906428674902800100, −6.66673745742666240313818091326, −5.53058080250058724906777399340, −4.52219498634741231669669679079, −3.80311146636830793196719386166, −3.20657567835100196539480935947, −1.70318741793399890171852194183, −0.74995212947272138731747924491, 0.74995212947272138731747924491, 1.70318741793399890171852194183, 3.20657567835100196539480935947, 3.80311146636830793196719386166, 4.52219498634741231669669679079, 5.53058080250058724906777399340, 6.66673745742666240313818091326, 7.34419981263906428674902800100, 7.912467067480556210729300376728, 8.705840044240023545250390625473

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