Properties

Label 2-3549-1.1-c1-0-39
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  − 0.374·2-s − 3-s − 1.85·4-s + 1.32·5-s + 0.374·6-s − 7-s + 1.44·8-s + 9-s − 0.497·10-s + 5.82·11-s + 1.85·12-s + 0.374·14-s − 1.32·15-s + 3.17·16-s − 0.442·17-s − 0.374·18-s + 1.34·19-s − 2.46·20-s + 21-s − 2.18·22-s − 2.67·23-s − 1.44·24-s − 3.23·25-s − 27-s + 1.85·28-s + 3.35·29-s + 0.497·30-s + ⋯
L(s)  = 1  − 0.265·2-s − 0.577·3-s − 0.929·4-s + 0.593·5-s + 0.153·6-s − 0.377·7-s + 0.511·8-s + 0.333·9-s − 0.157·10-s + 1.75·11-s + 0.536·12-s + 0.100·14-s − 0.342·15-s + 0.794·16-s − 0.107·17-s − 0.0883·18-s + 0.308·19-s − 0.551·20-s + 0.218·21-s − 0.465·22-s − 0.558·23-s − 0.295·24-s − 0.647·25-s − 0.192·27-s + 0.351·28-s + 0.622·29-s + 0.0908·30-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

\(L(1)\) \(\approx\) \(1.254181915\)
\(L(\frac12)\) \(\approx\) \(1.254181915\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.374T + 2T^{2} \)
5 \( 1 - 1.32T + 5T^{2} \)
11 \( 1 - 5.82T + 11T^{2} \)
17 \( 1 + 0.442T + 17T^{2} \)
19 \( 1 - 1.34T + 19T^{2} \)
23 \( 1 + 2.67T + 23T^{2} \)
29 \( 1 - 3.35T + 29T^{2} \)
31 \( 1 - 8.32T + 31T^{2} \)
37 \( 1 - 7.93T + 37T^{2} \)
41 \( 1 + 0.816T + 41T^{2} \)
43 \( 1 - 3.16T + 43T^{2} \)
47 \( 1 - 4.29T + 47T^{2} \)
53 \( 1 + 1.85T + 53T^{2} \)
59 \( 1 + 6.16T + 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 1.64T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 - 7.20T + 73T^{2} \)
79 \( 1 + 2.54T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 + 0.249T + 89T^{2} \)
97 \( 1 + 3.75T + 97T^{2} \)
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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

−8.747676520286840728176733087522, −7.908145348994592486942689705429, −6.99614335106171504355424078037, −6.10836997515833583635779417407, −5.83804224139274646538345047351, −4.52882064205434899349812015403, −4.23155324313815129778177037242, −3.10456887296203496877972712263, −1.65366928945394174546471427719, −0.76353995893802135008619710114, 0.76353995893802135008619710114, 1.65366928945394174546471427719, 3.10456887296203496877972712263, 4.23155324313815129778177037242, 4.52882064205434899349812015403, 5.83804224139274646538345047351, 6.10836997515833583635779417407, 6.99614335106171504355424078037, 7.908145348994592486942689705429, 8.747676520286840728176733087522

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