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Properties

Label 1-3549-3549.2-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.985 + 0.170i$
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  + (0.464 + 0.885i)2-s + (−0.568 + 0.822i)4-s + (0.160 + 0.987i)5-s + (−0.992 − 0.120i)8-s + (−0.799 + 0.600i)10-s + (−0.999 + 0.0402i)11-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + (0.866 + 0.5i)19-s + (−0.903 − 0.428i)20-s + (−0.5 − 0.866i)22-s + 23-s + (−0.948 + 0.316i)25-s + (0.0402 − 0.999i)29-s + (−0.979 + 0.200i)31-s + (0.663 − 0.748i)32-s + ⋯
L(s)  = 1  + (0.464 + 0.885i)2-s + (−0.568 + 0.822i)4-s + (0.160 + 0.987i)5-s + (−0.992 − 0.120i)8-s + (−0.799 + 0.600i)10-s + (−0.999 + 0.0402i)11-s + (−0.354 − 0.935i)16-s + (0.120 − 0.992i)17-s + (0.866 + 0.5i)19-s + (−0.903 − 0.428i)20-s + (−0.5 − 0.866i)22-s + 23-s + (−0.948 + 0.316i)25-s + (0.0402 − 0.999i)29-s + (−0.979 + 0.200i)31-s + (0.663 − 0.748i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.985 + 0.170i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.985 + 0.170i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.159980236 + 0.09989559744i\)
\(L(\frac12)\) \(\approx\) \(1.159980236 + 0.09989559744i\)
\(L(1)\) \(\approx\) \(0.9187045335 + 0.5687099755i\)
\(L(1)\) \(\approx\) \(0.9187045335 + 0.5687099755i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.464 + 0.885i)T \)
5 \( 1 + (0.160 + 0.987i)T \)
11 \( 1 + (-0.999 + 0.0402i)T \)
17 \( 1 + (0.120 - 0.992i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + T \)
29 \( 1 + (0.0402 - 0.999i)T \)
31 \( 1 + (-0.979 + 0.200i)T \)
37 \( 1 + (-0.663 - 0.748i)T \)
41 \( 1 + (0.960 + 0.278i)T \)
43 \( 1 + (0.200 - 0.979i)T \)
47 \( 1 + (-0.903 - 0.428i)T \)
53 \( 1 + (-0.799 - 0.600i)T \)
59 \( 1 + (0.935 + 0.354i)T \)
61 \( 1 + (-0.919 - 0.391i)T \)
67 \( 1 + (-0.903 - 0.428i)T \)
71 \( 1 + (-0.721 + 0.692i)T \)
73 \( 1 + (-0.534 - 0.845i)T \)
79 \( 1 + (0.428 - 0.903i)T \)
83 \( 1 + (0.239 - 0.970i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.774 + 0.632i)T \)
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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

−18.864934459962559837762696873910, −18.03436702791092544265831371006, −17.55056696446876823311398980785, −16.60301550316300855833328383634, −15.905472685063413197826273006269, −15.18508038083952618557933401999, −14.42675396561359112885890323425, −13.62520324323375135487292858338, −12.85674379073861157878778211562, −12.77761477745145408616066710063, −11.825997073029752682485190492570, −11.02383977111946231689561203378, −10.48224606443818774487223413598, −9.59396197748310893702492335564, −9.05355925504590951224653234266, −8.3165645501257330509920875544, −7.46968398337407107499995963729, −6.31447653465275167500332245640, −5.44842077844847957429297676981, −5.05418181963202910152379873070, −4.29728878162556604931873448246, −3.34867170297857161071132376185, −2.65977205739739302704145358634, −1.61112692887936675640913009583, −1.01961864272334954585573153366, 0.30977936960765685480522206309, 2.00028614718368227646597853053, 2.98337518820964206737076486074, 3.356712012856419596798807032831, 4.45533822371922583484402275484, 5.31660547503558869744839714407, 5.78574480853788632360379481433, 6.70517171447238462665919971961, 7.48578948794798847039505843004, 7.657762357758325425264224874015, 8.8204521382708426069970399246, 9.55063645775765385920233890288, 10.29382100382382237337438132101, 11.16006745653602888123563148342, 11.84720397001518854647467700296, 12.70580295247642721811354775965, 13.47655122999242801814861803076, 13.925750001470821438951262922223, 14.71401076519982368740160437163, 15.17768600464714412467451199908, 16.08421380704299614106432710091, 16.32046804175580909993866733622, 17.60685022476252432338647677349, 17.79433750748097652617825300174, 18.675198930476905013338547715041

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