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Properties

Label 1-3549-3549.2183-r0-0-0
Degree $1$
Conductor $3549$
Sign $-0.230 + 0.973i$
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  + (−0.354 − 0.935i)2-s + (−0.748 + 0.663i)4-s + (−0.120 − 0.992i)5-s + (0.885 + 0.464i)8-s + (−0.885 + 0.464i)10-s + (−0.354 + 0.935i)11-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + 19-s + (0.748 + 0.663i)20-s + 22-s − 23-s + (−0.970 + 0.239i)25-s + (0.354 + 0.935i)29-s + (−0.970 − 0.239i)31-s + (−0.970 + 0.239i)32-s + ⋯
L(s)  = 1  + (−0.354 − 0.935i)2-s + (−0.748 + 0.663i)4-s + (−0.120 − 0.992i)5-s + (0.885 + 0.464i)8-s + (−0.885 + 0.464i)10-s + (−0.354 + 0.935i)11-s + (0.120 − 0.992i)16-s + (0.885 + 0.464i)17-s + 19-s + (0.748 + 0.663i)20-s + 22-s − 23-s + (−0.970 + 0.239i)25-s + (0.354 + 0.935i)29-s + (−0.970 − 0.239i)31-s + (−0.970 + 0.239i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $-0.230 + 0.973i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (2183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ -0.230 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07430260240 + 0.09393772312i\)
\(L(\frac12)\) \(\approx\) \(0.07430260240 + 0.09393772312i\)
\(L(1)\) \(\approx\) \(0.6275743619 - 0.2881712404i\)
\(L(1)\) \(\approx\) \(0.6275743619 - 0.2881712404i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.354 - 0.935i)T \)
5 \( 1 + (-0.120 - 0.992i)T \)
11 \( 1 + (-0.354 + 0.935i)T \)
17 \( 1 + (0.885 + 0.464i)T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + (0.354 + 0.935i)T \)
31 \( 1 + (-0.970 - 0.239i)T \)
37 \( 1 + (0.970 + 0.239i)T \)
41 \( 1 + (-0.568 + 0.822i)T \)
43 \( 1 + (-0.970 + 0.239i)T \)
47 \( 1 + (0.748 + 0.663i)T \)
53 \( 1 + (-0.885 - 0.464i)T \)
59 \( 1 + (-0.120 - 0.992i)T \)
61 \( 1 + (-0.885 + 0.464i)T \)
67 \( 1 + (0.748 + 0.663i)T \)
71 \( 1 + (0.568 - 0.822i)T \)
73 \( 1 + (-0.354 + 0.935i)T \)
79 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (-0.568 - 0.822i)T \)
89 \( 1 - T \)
97 \( 1 + (0.120 - 0.992i)T \)
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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

−18.35410497354799928273350941483, −18.08678042547319738537645708758, −17.01540604820661912894052237150, −16.46027386368158499680447773990, −15.70820580208646362877445446815, −15.2931166650431008873130508330, −14.25772423673061033966432096324, −14.01982690528304031419840877340, −13.377428236596736646392626256647, −12.21960422397062866127973289597, −11.43791395363383817451757935646, −10.6809261901022333936985036444, −10.00569727336391109230132421939, −9.43771241140039662432539408054, −8.43790037421276000044872337023, −7.76397141704237928407615220175, −7.30397490557413649194502505740, −6.41180618787031245395521649227, −5.762067845560580638995222664, −5.20186604967385994666062255289, −4.00224824924535221399234313594, −3.353852094992691499106376581759, −2.39799927408061995757711903156, −1.18076280884196061792366509717, −0.04317875951241562951959827353, 1.28867876961660199462132734557, 1.68965863217948945442363307678, 2.817709826613124425359884182529, 3.63057758676249133736308846767, 4.43968859418727125215367962406, 5.06191720144758251258579901611, 5.807612056580058225323248443460, 7.15108203637410707112076089281, 7.916313785193392821066183697266, 8.31589216326788242826940428137, 9.38049985671688737648115908801, 9.727003230608542242097950891811, 10.41841647713567640108137551039, 11.40946178332929042339427839857, 11.98399102449448010752465589861, 12.659671250992383062408140778489, 13.0093170552932989857478475993, 13.94957532674505451161538275585, 14.62535327979414880572179661126, 15.67200668324643268190709215458, 16.39013869164557680997752016674, 16.896842727545800923466947461088, 17.73113805004449262219291348531, 18.234560397901050370516337082394, 18.92417577606632498357276782476

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