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Properties

Label 2-3549-3549.2693-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: $\chi_{3549} (2693, \cdot )$
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

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

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