Properties

Label 2-3549-3549.2669-c0-0-0
Degree $2$
Conductor $3549$
Sign $0.999 - 0.0354i$
Analytic cond. $1.77118$
Root an. cond. $1.33085$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.120 + 0.992i)3-s + (0.845 − 0.534i)4-s + (0.948 + 0.316i)7-s + (−0.970 − 0.239i)9-s + (0.428 + 0.903i)12-s + (0.5 − 0.866i)13-s + (0.428 − 0.903i)16-s − 1.92i·19-s + (−0.428 + 0.903i)21-s + (0.632 − 0.774i)25-s + (0.354 − 0.935i)27-s + (0.970 − 0.239i)28-s + (−0.149 − 0.917i)31-s + (−0.948 + 0.316i)36-s + (0.101 + 0.625i)37-s + ⋯
L(s)  = 1  + (−0.120 + 0.992i)3-s + (0.845 − 0.534i)4-s + (0.948 + 0.316i)7-s + (−0.970 − 0.239i)9-s + (0.428 + 0.903i)12-s + (0.5 − 0.866i)13-s + (0.428 − 0.903i)16-s − 1.92i·19-s + (−0.428 + 0.903i)21-s + (0.632 − 0.774i)25-s + (0.354 − 0.935i)27-s + (0.970 − 0.239i)28-s + (−0.149 − 0.917i)31-s + (−0.948 + 0.316i)36-s + (0.101 + 0.625i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.999 - 0.0354i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2669, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ 0.999 - 0.0354i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.666223885\)
\(L(\frac12)\) \(\approx\) \(1.666223885\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.120 - 0.992i)T \)
7 \( 1 + (-0.948 - 0.316i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.845 + 0.534i)T^{2} \)
5 \( 1 + (-0.632 + 0.774i)T^{2} \)
11 \( 1 + (0.885 - 0.464i)T^{2} \)
17 \( 1 + (-0.799 + 0.600i)T^{2} \)
19 \( 1 + 1.92iT - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.0402 + 0.999i)T^{2} \)
31 \( 1 + (0.149 + 0.917i)T + (-0.948 + 0.316i)T^{2} \)
37 \( 1 + (-0.101 - 0.625i)T + (-0.948 + 0.316i)T^{2} \)
41 \( 1 + (0.278 + 0.960i)T^{2} \)
43 \( 1 + (1.19 - 1.46i)T + (-0.200 - 0.979i)T^{2} \)
47 \( 1 + (-0.996 - 0.0804i)T^{2} \)
53 \( 1 + (0.919 + 0.391i)T^{2} \)
59 \( 1 + (-0.632 + 0.774i)T^{2} \)
61 \( 1 + (1.26 - 1.12i)T + (0.120 - 0.992i)T^{2} \)
67 \( 1 + (-0.869 - 1.65i)T + (-0.568 + 0.822i)T^{2} \)
71 \( 1 + (0.692 - 0.721i)T^{2} \)
73 \( 1 + (1.30 - 1.25i)T + (0.0402 - 0.999i)T^{2} \)
79 \( 1 + (-0.00970 + 0.240i)T + (-0.996 - 0.0804i)T^{2} \)
83 \( 1 + (-0.970 + 0.239i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.56 + 0.742i)T + (0.632 + 0.774i)T^{2} \)
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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

−8.727137746718792339391586201364, −8.175676248726794992426004991529, −7.23307458869576090059976821721, −6.34794759435236482859979718142, −5.66331420212818014987095002264, −4.96735720197176233720401144382, −4.36269950662993969655358619602, −3.01845584447451220058362237822, −2.50221431495285931156813314182, −1.06475335873097170054927313640, 1.61283309155761557172944988858, 1.79427568790627461487621470649, 3.17107142945812954185343984050, 3.91403574943778683189965622271, 5.12222192522003649642108331549, 5.92858990460274020511070805603, 6.69659553741038756144645434524, 7.23965359843709753387492946026, 7.972034728507066787306825398538, 8.378406476425640878663609680570

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