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Properties

Label 2-3549-3549.2243-c0-0-0
Degree $2$
Conductor $3549$
Sign $-0.819 - 0.573i$
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  + (0.600 + 0.799i)3-s + (0.464 + 0.885i)4-s + (−0.903 − 0.428i)7-s + (−0.278 + 0.960i)9-s + (−0.428 + 0.903i)12-s + (−0.866 − 0.5i)13-s + (−0.568 + 0.822i)16-s + (0.517 + 1.93i)19-s + (−0.200 − 0.979i)21-s + (−0.774 + 0.632i)25-s + (−0.935 + 0.354i)27-s + (−0.0402 − 0.999i)28-s + (0.482 − 0.347i)31-s + (−0.979 + 0.200i)36-s + (−1.52 − 0.684i)37-s + ⋯
L(s)  = 1  + (0.600 + 0.799i)3-s + (0.464 + 0.885i)4-s + (−0.903 − 0.428i)7-s + (−0.278 + 0.960i)9-s + (−0.428 + 0.903i)12-s + (−0.866 − 0.5i)13-s + (−0.568 + 0.822i)16-s + (0.517 + 1.93i)19-s + (−0.200 − 0.979i)21-s + (−0.774 + 0.632i)25-s + (−0.935 + 0.354i)27-s + (−0.0402 − 0.999i)28-s + (0.482 − 0.347i)31-s + (−0.979 + 0.200i)36-s + (−1.52 − 0.684i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.819 - 0.573i$
Analytic conductor: \(1.77118\)
Root analytic conductor: \(1.33085\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3549,\ (\ :0),\ -0.819 - 0.573i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.232624524\)
\(L(\frac12)\) \(\approx\) \(1.232624524\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.600 - 0.799i)T \)
7 \( 1 + (0.903 + 0.428i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
good2 \( 1 + (-0.464 - 0.885i)T^{2} \)
5 \( 1 + (0.774 - 0.632i)T^{2} \)
11 \( 1 + (-0.534 - 0.845i)T^{2} \)
17 \( 1 + (-0.120 + 0.992i)T^{2} \)
19 \( 1 + (-0.517 - 1.93i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.845 + 0.534i)T^{2} \)
31 \( 1 + (-0.482 + 0.347i)T + (0.316 - 0.948i)T^{2} \)
37 \( 1 + (1.52 + 0.684i)T + (0.663 + 0.748i)T^{2} \)
41 \( 1 + (-0.721 + 0.692i)T^{2} \)
43 \( 1 + (-0.314 - 1.93i)T + (-0.948 + 0.316i)T^{2} \)
47 \( 1 + (0.0804 + 0.996i)T^{2} \)
53 \( 1 + (0.919 - 0.391i)T^{2} \)
59 \( 1 + (0.935 + 0.354i)T^{2} \)
61 \( 1 + (0.0255 + 0.0763i)T + (-0.799 + 0.600i)T^{2} \)
67 \( 1 + (-1.23 + 1.34i)T + (-0.0804 - 0.996i)T^{2} \)
71 \( 1 + (-0.960 - 0.278i)T^{2} \)
73 \( 1 + (-1.29 + 0.0260i)T + (0.999 - 0.0402i)T^{2} \)
79 \( 1 + (-0.0799 - 1.98i)T + (-0.996 + 0.0804i)T^{2} \)
83 \( 1 + (0.239 - 0.970i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-0.394 + 0.335i)T + (0.160 - 0.987i)T^{2} \)
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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

−9.147780553627854490811047272950, −7.997602740363258049750374407929, −7.87082227729385345296994161282, −7.00655544209656570087608400144, −6.06446477083708310072381483238, −5.21304568874099841627390664872, −4.09892990923850345234949141140, −3.54798514425555970184572228992, −2.94740258329985071651994663668, −1.93998045390107658601096111234, 0.60372786407560378620309318454, 2.05114764092007291046139328550, 2.55966533155678605848498061865, 3.47696354847503636607704824934, 4.78802740850422346523971478264, 5.56428313079461909669515375360, 6.47147549800510369728378406869, 6.91090816728835623885893294760, 7.42063624257547318678377411609, 8.635504606961462200455258739676

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