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Properties

Label 1-3549-3549.272-r0-0-0
Degree $1$
Conductor $3549$
Sign $0.848 + 0.529i$
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 + (−0.568 − 0.822i)5-s + (−0.748 + 0.663i)8-s + (0.748 + 0.663i)10-s + (−0.970 − 0.239i)11-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s + 19-s + (−0.885 − 0.464i)20-s + 22-s − 23-s + (−0.354 + 0.935i)25-s + (0.970 − 0.239i)29-s + (−0.354 − 0.935i)31-s + (−0.354 + 0.935i)32-s + ⋯
L(s)  = 1  + (−0.970 + 0.239i)2-s + (0.885 − 0.464i)4-s + (−0.568 − 0.822i)5-s + (−0.748 + 0.663i)8-s + (0.748 + 0.663i)10-s + (−0.970 − 0.239i)11-s + (0.568 − 0.822i)16-s + (−0.748 + 0.663i)17-s + 19-s + (−0.885 − 0.464i)20-s + 22-s − 23-s + (−0.354 + 0.935i)25-s + (0.970 − 0.239i)29-s + (−0.354 − 0.935i)31-s + (−0.354 + 0.935i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3549\)    =    \(3 \cdot 7 \cdot 13^{2}\)
Sign: $0.848 + 0.529i$
Analytic conductor: \(16.4814\)
Root analytic conductor: \(16.4814\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3549} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3549,\ (0:\ ),\ 0.848 + 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5193960784 + 0.1487022543i\)
\(L(\frac12)\) \(\approx\) \(0.5193960784 + 0.1487022543i\)
\(L(1)\) \(\approx\) \(0.5446810210 + 0.01906060271i\)
\(L(1)\) \(\approx\) \(0.5446810210 + 0.01906060271i\)

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 \( 1 + (-0.568 - 0.822i)T \)
11 \( 1 + (-0.970 - 0.239i)T \)
17 \( 1 + (-0.748 + 0.663i)T \)
19 \( 1 + T \)
23 \( 1 - T \)
29 \( 1 + (0.970 - 0.239i)T \)
31 \( 1 + (-0.354 - 0.935i)T \)
37 \( 1 + (0.354 + 0.935i)T \)
41 \( 1 + (-0.120 - 0.992i)T \)
43 \( 1 + (-0.354 + 0.935i)T \)
47 \( 1 + (-0.885 - 0.464i)T \)
53 \( 1 + (0.748 - 0.663i)T \)
59 \( 1 + (-0.568 - 0.822i)T \)
61 \( 1 + (0.748 + 0.663i)T \)
67 \( 1 + (-0.885 - 0.464i)T \)
71 \( 1 + (0.120 + 0.992i)T \)
73 \( 1 + (-0.970 - 0.239i)T \)
79 \( 1 + (0.885 + 0.464i)T \)
83 \( 1 + (-0.120 + 0.992i)T \)
89 \( 1 - T \)
97 \( 1 + (0.568 - 0.822i)T \)
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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

−18.46238575991967120309222332418, −17.99913491858977581340578902154, −17.76669236490744848093476832139, −16.43824107198268382542430613001, −15.960584050627984912708707980886, −15.53350452780012985254833000157, −14.68128460843356658115272404059, −13.87161674124764612189995074906, −13.00769747847278428140420331871, −12.03186654334812497801746192387, −11.69428652010949477670011185140, −10.76553180867425534623841393057, −10.38450054635169525069054559571, −9.63119331936388200720878574691, −8.79769616129071684793990503756, −7.9804433342633095595360227205, −7.45611092833112198073124030416, −6.85052980883685091561630207361, −6.04921665639145108027679701781, −4.99413256186008208940857674344, −3.98299050131871375252901293959, −3.0273795029329700924039246859, −2.611654340916626304136438340269, −1.62175025245703820150227578472, −0.350115723361912243716416075854, 0.61751889665200400493491941823, 1.594958676840148747545889277159, 2.4693459939475532875062741360, 3.43405015835412876516182123368, 4.451962994564863970592907829671, 5.29764295652986302434090914225, 5.96338994748756682976425879819, 6.87207010927213972394351890157, 7.74895630850270433492083542598, 8.21119059656411690332167844883, 8.73215898806840887082295372257, 9.712064221724057480310796266426, 10.16357988112443652340214347355, 11.15971991809061335090644599624, 11.63288956295927210982920587340, 12.41092469264290403639660054380, 13.21059018615792828022361040610, 13.94466561238466421898551236582, 15.08743454745115633487238433648, 15.47410296019435085587767302008, 16.24429595319394311916561259444, 16.53537949055826998903870913641, 17.52932522827637372997953278030, 18.01098908013591141410289167726, 18.7342080957312273046817416149

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