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Properties

Label 2-39-39.20-c1-0-0
Degree $2$
Conductor $39$
Sign $0.248 - 0.968i$
Analytic cond. $0.311416$
Root an. cond. $0.558047$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.619 + 2.31i)2-s + (−1.28 − 1.16i)3-s + (−3.23 + 1.86i)4-s + (1.69 − 1.69i)5-s + (1.89 − 3.68i)6-s + (−1.36 − 0.366i)7-s + (−2.93 − 2.93i)8-s + (0.292 + 2.98i)9-s + (4.96 + 2.86i)10-s + (−1.69 + 0.453i)11-s + (6.31 + 1.36i)12-s + (−1.59 − 3.23i)13-s − 3.38i·14-s + (−4.14 + 0.202i)15-s + (1.23 − 2.13i)16-s + (1.07 + 1.85i)17-s + ⋯
L(s)  = 1  + (0.438 + 1.63i)2-s + (−0.740 − 0.671i)3-s + (−1.61 + 0.933i)4-s + (0.757 − 0.757i)5-s + (0.773 − 1.50i)6-s + (−0.516 − 0.138i)7-s + (−1.03 − 1.03i)8-s + (0.0975 + 0.995i)9-s + (1.56 + 0.906i)10-s + (−0.510 + 0.136i)11-s + (1.82 + 0.394i)12-s + (−0.443 − 0.896i)13-s − 0.904i·14-s + (−1.06 + 0.0523i)15-s + (0.308 − 0.533i)16-s + (0.260 + 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.248 - 0.968i$
Analytic conductor: \(0.311416\)
Root analytic conductor: \(0.558047\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :1/2),\ 0.248 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.603367 + 0.467915i\)
\(L(\frac12)\) \(\approx\) \(0.603367 + 0.467915i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.28 + 1.16i)T \)
13 \( 1 + (1.59 + 3.23i)T \)
good2 \( 1 + (-0.619 - 2.31i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (-1.69 + 1.69i)T - 5iT^{2} \)
7 \( 1 + (1.36 + 0.366i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.69 - 0.453i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.07 - 1.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.267 - i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.79 - 2.76i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.46 - 4.46i)T + 31iT^{2} \)
37 \( 1 + (1.76 + 6.59i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.166 + 0.619i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-7.09 + 4.09i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.77 + 6.77i)T + 47iT^{2} \)
53 \( 1 - 4.62iT - 53T^{2} \)
59 \( 1 + (-1.23 + 4.62i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.46 - 2.26i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.62 - 1.23i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (6.09 - 6.09i)T - 73iT^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + (1.23 - 1.23i)T - 83iT^{2} \)
89 \( 1 + (9.70 - 2.60i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-3.36 + 12.5i)T + (-84.0 - 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.54653774194307208799596870522, −15.66317440429096813659552138770, −14.16084729970142692144691609618, −13.10007655399432880274294286126, −12.52839326432450002874148299896, −10.24220840033534400824433356272, −8.461159458642994795156395746183, −7.19487632050671802615762527029, −5.91572767614074564480228031070, −5.04852373394208917471091899464, 2.85080616848113840615003351245, 4.66855553541343769555966900722, 6.32027748736098314606872610873, 9.527678998635192552840990833118, 10.12230220728853181358322335186, 11.20488791894176410864793618509, 12.15203152923105958253026705160, 13.44072305692369136671791587121, 14.51173279056174674504263381431, 16.04718524146681882632116795961

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