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Properties

Label 1-3549-3549.419-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.956 + 0.293i$
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  + (−0.428 − 0.903i)2-s + (−0.632 + 0.774i)4-s + (−0.748 − 0.663i)5-s + (0.970 + 0.239i)8-s + (−0.278 + 0.960i)10-s + (0.996 − 0.0804i)11-s + (−0.200 − 0.979i)16-s + (0.278 + 0.960i)17-s + (0.5 + 0.866i)19-s + (0.987 − 0.160i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (0.120 + 0.992i)25-s + (−0.428 − 0.903i)29-s + (−0.120 + 0.992i)31-s + (−0.799 + 0.600i)32-s + ⋯
L(s)  = 1  + (−0.428 − 0.903i)2-s + (−0.632 + 0.774i)4-s + (−0.748 − 0.663i)5-s + (0.970 + 0.239i)8-s + (−0.278 + 0.960i)10-s + (0.996 − 0.0804i)11-s + (−0.200 − 0.979i)16-s + (0.278 + 0.960i)17-s + (0.5 + 0.866i)19-s + (0.987 − 0.160i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + (0.120 + 0.992i)25-s + (−0.428 − 0.903i)29-s + (−0.120 + 0.992i)31-s + (−0.799 + 0.600i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.956 + 0.293i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.956 + 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9192141561 + 0.1377171736i\)
\(L(\frac12)\) \(\approx\) \(0.9192141561 + 0.1377171736i\)
\(L(1)\) \(\approx\) \(0.7228963917 - 0.2371716042i\)
\(L(1)\) \(\approx\) \(0.7228963917 - 0.2371716042i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.428 - 0.903i)T \)
5 \( 1 + (-0.748 - 0.663i)T \)
11 \( 1 + (0.996 - 0.0804i)T \)
17 \( 1 + (0.278 + 0.960i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.428 - 0.903i)T \)
31 \( 1 + (-0.120 + 0.992i)T \)
37 \( 1 + (0.799 + 0.600i)T \)
41 \( 1 + (-0.0402 + 0.999i)T \)
43 \( 1 + (0.799 - 0.600i)T \)
47 \( 1 + (-0.354 + 0.935i)T \)
53 \( 1 + (0.970 + 0.239i)T \)
59 \( 1 + (-0.200 + 0.979i)T \)
61 \( 1 + (-0.692 - 0.721i)T \)
67 \( 1 + (-0.632 - 0.774i)T \)
71 \( 1 + (0.845 + 0.534i)T \)
73 \( 1 + (-0.568 - 0.822i)T \)
79 \( 1 + (-0.354 + 0.935i)T \)
83 \( 1 + (0.885 + 0.464i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.948 - 0.316i)T \)
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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

−18.64358896507300486886653614086, −17.91221401330232791198047789904, −17.39941842098598343622002729100, −16.40211196206374844113550758729, −16.11317755803645848390420646418, −15.08719772615747997448814252538, −14.89077689726826095722392085168, −14.01423762330949829965721928786, −13.47712755252816415105669524296, −12.432812609229347380536079813809, −11.44832960554599838362469014054, −11.154241003520237430841742084093, −10.12407774311756439622341765059, −9.33076863301697869941179127024, −8.92419949801285670902557360014, −7.8694370714246028933380498557, −7.2049423507549289119839775155, −6.93902441827769778886019947239, −5.93780671811024821139042883423, −5.18429680124374423857448412799, −4.28703985555362547308999522308, −3.605304433869786402353541181917, −2.61600994056731985992051131443, −1.36798172097723651833204578968, −0.39772311391906942441572760681, 1.041222024236805158826347618121, 1.43753196830996505996696466213, 2.65269367443912625383223395845, 3.57734604010060221763142761233, 4.09223635438904420454803392058, 4.77525240906348953809021210945, 5.80686969886015411923023312188, 6.79993878102265330185982680289, 7.82935803390655672074008118, 8.19723354824344793686686356706, 9.03619148076787896803137185746, 9.54038409365960476541582955277, 10.46840863873499769282380594391, 11.11889452337506281365332819551, 11.906421961853468406576767606339, 12.306802284277095895067519321128, 12.9298973883802562260894660365, 13.77833968651715628548930540742, 14.57735454501137336283398158065, 15.29215255482979289448808514187, 16.446911675273013779269782900555, 16.6701200323049096818802406974, 17.342716215238627325911999973117, 18.244061171384082052876250268407, 18.937444471936092301514296437026

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