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Properties

Label 1-39-39.11-r0-0-0
Degree $1$
Conductor $39$
Sign $0.846 + 0.533i$
Analytic cond. $0.181115$
Root an. cond. $0.181115$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s − i·5-s + (−0.866 + 0.5i)7-s + i·8-s + (0.5 − 0.866i)10-s + (−0.866 − 0.5i)11-s − 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s − 25-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s − i·5-s + (−0.866 + 0.5i)7-s + i·8-s + (0.5 − 0.866i)10-s + (−0.866 − 0.5i)11-s − 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)19-s + (0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s − 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.846 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.846 + 0.533i$
Analytic conductor: \(0.181115\)
Root analytic conductor: \(0.181115\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 39,\ (0:\ ),\ 0.846 + 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.091680137 + 0.3152253091i\)
\(L(\frac12)\) \(\approx\) \(1.091680137 + 0.3152253091i\)
\(L(1)\) \(\approx\) \(1.309352061 + 0.2938183146i\)
\(L(1)\) \(\approx\) \(1.309352061 + 0.2938183146i\)

Euler product

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

−34.94840091752097655032284348874, −33.641631781691073516687761468761, −32.8067485055861462375206753926, −31.41517711643276767706869314577, −30.476866884860633835685338017965, −29.39259922889505116447657922061, −28.45642092789519855145870402041, −26.64520102730232883890401700039, −25.571480123759350791174828221449, −23.875321898851921749626352460092, −22.81735441628369306586682238299, −22.0337716991536068719830566479, −20.54007140588074506085714090267, −19.38671812014595237895865606321, −18.19218586998832770660884922619, −16.099699350134207202116911708833, −14.89578210679081190352577426472, −13.64979701360969999738762670439, −12.46203608168053526366124982173, −10.80338999148904019345750463890, −9.96135721152985103957219737700, −7.25208196541185733982899051650, −5.96716075357492301283651387094, −3.965856835165729107788502321, −2.550655005150280048324763697161, 2.932167245890278770202944209418, 4.81661252838882842811719405400, 6.03952750272900733390850490632, 7.82722605729856712753648415116, 9.32609519612752346890052481788, 11.60434732444579678329844912919, 12.84783644830002292148204968312, 13.72838440997076801374861968415, 15.74248807801981662824813660975, 16.15001454852903613804622847343, 17.79668449184314334782532395118, 19.7159259103815814671214822799, 20.96671903039801811739904519355, 22.07098332404063054473224248886, 23.37275821536238455608460168965, 24.465319051404606752004739202645, 25.41511926162147299081067111569, 26.70910401010646630334960340647, 28.56323721981635983341897288854, 29.3747347001883007877640848831, 31.10572539594234415118851424323, 31.917297154864845213487568002821, 32.73349609538945884017253872803, 34.09009108560918416031322091800, 35.24866494393519098847528364046

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