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Properties

Label 1-3549-3549.269-r0-0-0
Degree $1$
Conductor $3549$
Sign $-0.774 - 0.632i$
Analytic cond. $16.4814$
Root an. cond. $16.4814$
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  + (0.970 − 0.239i)2-s + (0.885 − 0.464i)4-s + (−0.996 + 0.0804i)5-s + (0.748 − 0.663i)8-s + (−0.948 + 0.316i)10-s + (−0.692 + 0.721i)11-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s + (0.5 + 0.866i)19-s + (−0.845 + 0.534i)20-s + (−0.5 + 0.866i)22-s − 23-s + (0.987 − 0.160i)25-s + (−0.692 − 0.721i)29-s + (0.632 − 0.774i)31-s + (0.354 − 0.935i)32-s + ⋯
L(s)  = 1  + (0.970 − 0.239i)2-s + (0.885 − 0.464i)4-s + (−0.996 + 0.0804i)5-s + (0.748 − 0.663i)8-s + (−0.948 + 0.316i)10-s + (−0.692 + 0.721i)11-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s + (0.5 + 0.866i)19-s + (−0.845 + 0.534i)20-s + (−0.5 + 0.866i)22-s − 23-s + (0.987 − 0.160i)25-s + (−0.692 − 0.721i)29-s + (0.632 − 0.774i)31-s + (0.354 − 0.935i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.774 - 0.632i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ -0.774 - 0.632i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3845380441 - 1.079463342i\)
\(L(\frac12)\) \(\approx\) \(0.3845380441 - 1.079463342i\)
\(L(1)\) \(\approx\) \(1.282913278 - 0.2900756520i\)
\(L(1)\) \(\approx\) \(1.282913278 - 0.2900756520i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.970 - 0.239i)T \)
5 \( 1 + (-0.996 + 0.0804i)T \)
11 \( 1 + (-0.692 + 0.721i)T \)
17 \( 1 + (-0.748 + 0.663i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.692 - 0.721i)T \)
31 \( 1 + (0.632 - 0.774i)T \)
37 \( 1 + (-0.354 - 0.935i)T \)
41 \( 1 + (0.799 - 0.600i)T \)
43 \( 1 + (-0.632 - 0.774i)T \)
47 \( 1 + (-0.845 + 0.534i)T \)
53 \( 1 + (-0.948 - 0.316i)T \)
59 \( 1 + (0.568 + 0.822i)T \)
61 \( 1 + (0.200 - 0.979i)T \)
67 \( 1 + (-0.845 + 0.534i)T \)
71 \( 1 + (0.919 + 0.391i)T \)
73 \( 1 + (-0.278 - 0.960i)T \)
79 \( 1 + (-0.845 + 0.534i)T \)
83 \( 1 + (0.120 - 0.992i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.428 - 0.903i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.15338248508012693613458596204, −18.27555385614645897025380452427, −17.59587424015443574620954013023, −16.48213281172637808105841696420, −16.09141904502490667060953192317, −15.60742250322933218712612032610, −14.92029178168882651310467852282, −14.11981445654890340739677459542, −13.44540785854257917064841571193, −12.88620272785422224377651221734, −12.051726767482685887927484056294, −11.396150539882240750363803929889, −11.03780358787355459631496754729, −10.08065517961904841335637470995, −8.91103500451962837793691695081, −8.19163098050358054906829666090, −7.60506011093658820914753802915, −6.8378566269168123879726429527, −6.18440393506878628314416105549, −5.01071619662005678189791873093, −4.83505676890198456506524499527, −3.76044487301053035206332881757, −3.12442451884736840061033965855, −2.46404792877002679098571668402, −1.17883680513824296365004038894, 0.22527572932199385260888476878, 1.68615975702172688383320419955, 2.3439373558192558969431676141, 3.3297006510432957248826090290, 4.06015652623608047961593126940, 4.50461153443902439675239738951, 5.48927546606800665048738452519, 6.16257233244762894877629303120, 7.0877911629896427891383249920, 7.7084774034482046050748086543, 8.29096378917283719666381491582, 9.56012142160482323808941671975, 10.272308236644947633626334694868, 10.94218001419918728759795907147, 11.6478729116915482930885232362, 12.25194027571446839832164580880, 12.83553720682106536789157664979, 13.52235480573377645217986271608, 14.43045307945932240069538196557, 14.93769090312545492850799318390, 15.74592824290220768353309609215, 15.93063032980908703667705250781, 16.918050678128906616896911397089, 17.82850580584430907919075429049, 18.68134375991559920581896979520

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