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Properties

Label 1-3549-3549.908-r1-0-0
Degree $1$
Conductor $3549$
Sign $0.105 - 0.994i$
Analytic cond. $381.393$
Root an. cond. $381.393$
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  + (−0.935 + 0.354i)2-s + (0.748 − 0.663i)4-s + (0.600 + 0.799i)5-s + (−0.464 + 0.885i)8-s + (−0.845 − 0.534i)10-s + (0.160 + 0.987i)11-s + (0.120 − 0.992i)16-s + (−0.885 − 0.464i)17-s + (0.866 + 0.5i)19-s + (0.979 + 0.200i)20-s + (−0.5 − 0.866i)22-s + 23-s + (−0.278 + 0.960i)25-s + (−0.987 − 0.160i)29-s + (−0.721 − 0.692i)31-s + (0.239 + 0.970i)32-s + ⋯
L(s)  = 1  + (−0.935 + 0.354i)2-s + (0.748 − 0.663i)4-s + (0.600 + 0.799i)5-s + (−0.464 + 0.885i)8-s + (−0.845 − 0.534i)10-s + (0.160 + 0.987i)11-s + (0.120 − 0.992i)16-s + (−0.885 − 0.464i)17-s + (0.866 + 0.5i)19-s + (0.979 + 0.200i)20-s + (−0.5 − 0.866i)22-s + 23-s + (−0.278 + 0.960i)25-s + (−0.987 − 0.160i)29-s + (−0.721 − 0.692i)31-s + (0.239 + 0.970i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.105 - 0.994i$
Analytic conductor: \(381.393\)
Root analytic conductor: \(381.393\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (908, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (1:\ ),\ 0.105 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1975961898 - 0.1777561527i\)
\(L(\frac12)\) \(\approx\) \(0.1975961898 - 0.1777561527i\)
\(L(1)\) \(\approx\) \(0.6725765164 + 0.2367584633i\)
\(L(1)\) \(\approx\) \(0.6725765164 + 0.2367584633i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.935 + 0.354i)T \)
5 \( 1 + (0.600 + 0.799i)T \)
11 \( 1 + (0.160 + 0.987i)T \)
17 \( 1 + (-0.885 - 0.464i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.987 - 0.160i)T \)
31 \( 1 + (-0.721 - 0.692i)T \)
37 \( 1 + (-0.239 + 0.970i)T \)
41 \( 1 + (-0.903 + 0.428i)T \)
43 \( 1 + (-0.692 - 0.721i)T \)
47 \( 1 + (0.979 + 0.200i)T \)
53 \( 1 + (0.845 - 0.534i)T \)
59 \( 1 + (0.992 - 0.120i)T \)
61 \( 1 + (0.0402 - 0.999i)T \)
67 \( 1 + (-0.979 - 0.200i)T \)
71 \( 1 + (-0.0804 + 0.996i)T \)
73 \( 1 + (0.774 - 0.632i)T \)
79 \( 1 + (-0.200 + 0.979i)T \)
83 \( 1 + (-0.822 + 0.568i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.391 + 0.919i)T \)
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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

−18.6358452489024725501084266867, −17.99168014083099662467736091937, −17.392462115948216820008839618105, −16.703999062539153598214372036889, −16.27750851569911212878064521986, −15.52523737281406959846344206836, −14.671531510940870738978936116711, −13.5233664885703059498188521465, −13.22349942025352468414913433631, −12.37235441242590022502273514206, −11.62466249629554274112394561710, −10.92942801748216082953955916719, −10.35091936951095536160101341509, −9.34773543343664306378499397617, −8.91969136875596692974367476079, −8.49889805451977272219109123370, −7.43530537066877317325829810541, −6.792014176532623273743454138252, −5.85052919331474284849248114369, −5.223179478050906959200721082821, −4.08691358799205592071941393709, −3.27308499989923740464861703022, −2.38552983569946042040874243820, −1.52492596533090022192358806465, −0.83627046352312638306701146689, 0.06168088158084771845151968356, 1.3367541936014674733988964592, 2.04003263859887523460635060352, 2.74529694838178857445803576866, 3.73954722710808035413566874877, 5.02482809993514080347022124793, 5.58772633011332125044806417874, 6.58489469721699138505795888456, 7.041025932117702669553552461467, 7.58866467706626513525688642972, 8.61403157439527139025338967079, 9.41096887129959346555224561303, 9.82459378958436013774942813403, 10.520691808381178648250801482545, 11.314627703832391727367194722083, 11.794299723301845010025016383, 12.93987361502412585716757979291, 13.69570198726885388574026694951, 14.45789166373934563079184948699, 15.190117623963627003849516945016, 15.436063215470403256490281574543, 16.601116409125672519773157310217, 17.084767852332812176438447369161, 17.75109523353687473648779344028, 18.439333629564888572313857406653

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