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Properties

Label 1-39-39.29-r1-0-0
Degree $1$
Conductor $39$
Sign $-0.964 - 0.265i$
Analytic cond. $4.19113$
Root an. cond. $4.19113$
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  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s − 5-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s − 14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + 25-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s − 5-s + (−0.5 − 0.866i)7-s − 8-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s − 14-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 − 0.866i)23-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.964 - 0.265i$
Analytic conductor: \(4.19113\)
Root analytic conductor: \(4.19113\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 39,\ (1:\ ),\ -0.964 - 0.265i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1530210638 - 1.134200370i\)
\(L(\frac12)\) \(\approx\) \(0.1530210638 - 1.134200370i\)
\(L(1)\) \(\approx\) \(0.7159775651 - 0.7068544717i\)
\(L(1)\) \(\approx\) \(0.7159775651 - 0.7068544717i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 - T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−35.32649137190696217836248283803, −34.37845829495399721362242405215, −33.19078475862957126084605114829, −31.74087759973395236194167649809, −31.306543675824903907900343689329, −29.90528087218729727498055296240, −28.025931658399276533143785884186, −27.06520194952063278001866669933, −25.595111517359411906726792771820, −24.760718367849189766816697101317, −23.2387380378985510973466272994, −22.61904346231773345134457117628, −21.13402927093792242097509495760, −19.44940485468687866416799572394, −18.10094580194111281481458740738, −16.49934222984526457118900768562, −15.50251631485104679261234991962, −14.50635637184671347507827734758, −12.673980278200327162706234883445, −11.81982183794714357769878223013, −9.35735429048402031675389285651, −7.92594272081486605002969075297, −6.58951161394890941308514719609, −4.88632247839689758338231368888, −3.305312400127323354832623735309, 0.651505726624621291010116007042, 3.2574658391024775758425551230, 4.4134169614179791024605223856, 6.518687995873537281905639201842, 8.53211163195525383549041441111, 10.32210452463543980358058193129, 11.430474164473536114482753147466, 12.7381539237995034263200292191, 14.02648482475902075700685326525, 15.42958102191265837041042841563, 16.996620052471072225174062008222, 19.04772417237991044154416764691, 19.57978582057908374612261028082, 20.88947628164787160123158622951, 22.32310887182134723467439768893, 23.33067769085437875263309496725, 24.257954291827903182101839163134, 26.41018699807143072789051020837, 27.397850961283629574105804276115, 28.58730816896096674794119741960, 29.91516433508462984812402227833, 30.65883179987856584138557940204, 32.108790759314301633438597058782, 32.68923423769577713106784117218, 34.499315465757290836080630141731

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