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Properties

Label 2-39-39.32-c5-0-13
Degree $2$
Conductor $39$
Sign $-0.386 + 0.922i$
Analytic cond. $6.25496$
Root an. cond. $2.50099$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.79 + 13.5i)3-s + (−27.7 + 16i)4-s + (66.1 − 247. i)7-s + (−121.5 − 210. i)9-s − 498. i·12-s + (−470. + 387.5i)13-s + (511. − 886. i)16-s + (−2.89e3 − 774. i)19-s + (2.81e3 + 2.81e3i)21-s − 3.12e3i·25-s + 3.78e3·27-s + (2.11e3 + 7.90e3i)28-s + (−3.81e3 + 3.81e3i)31-s + (6.73e3 + 3.88e3i)36-s + (−1.00e4 + 2.68e3i)37-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)3-s + (−0.866 + 0.5i)4-s + (0.510 − 1.90i)7-s + (−0.5 − 0.866i)9-s − 0.999i·12-s + (−0.771 + 0.635i)13-s + (0.499 − 0.866i)16-s + (−1.83 − 0.492i)19-s + (1.39 + 1.39i)21-s − i·25-s + 1.00·27-s + (0.510 + 1.90i)28-s + (−0.712 + 0.712i)31-s + (0.866 + 0.5i)36-s + (−1.20 + 0.322i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(6.25496\)
Root analytic conductor: \(2.50099\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :5/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.203315 - 0.305569i\)
\(L(\frac12)\) \(\approx\) \(0.203315 - 0.305569i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (7.79 - 13.5i)T \)
13 \( 1 + (470. - 387.5i)T \)
good2 \( 1 + (27.7 - 16i)T^{2} \)
5 \( 1 + 3.12e3iT^{2} \)
7 \( 1 + (-66.1 + 247. i)T + (-1.45e4 - 8.40e3i)T^{2} \)
11 \( 1 + (-1.39e5 + 8.05e4i)T^{2} \)
17 \( 1 + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (2.89e3 + 774. i)T + (2.14e6 + 1.23e6i)T^{2} \)
23 \( 1 + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (3.81e3 - 3.81e3i)T - 2.86e7iT^{2} \)
37 \( 1 + (1.00e4 - 2.68e3i)T + (6.00e7 - 3.46e7i)T^{2} \)
41 \( 1 + (1.00e8 - 5.79e7i)T^{2} \)
43 \( 1 + (-1.29e4 + 7.45e3i)T + (7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 - 2.29e8iT^{2} \)
53 \( 1 - 4.18e8T^{2} \)
59 \( 1 + (6.19e8 + 3.57e8i)T^{2} \)
61 \( 1 + (5.86e3 + 1.01e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.84e4 - 6.88e4i)T + (-1.16e9 + 6.75e8i)T^{2} \)
71 \( 1 + (-1.56e9 - 9.02e8i)T^{2} \)
73 \( 1 + (1.76e4 + 1.76e4i)T + 2.07e9iT^{2} \)
79 \( 1 - 6.36e4T + 3.07e9T^{2} \)
83 \( 1 + 3.93e9iT^{2} \)
89 \( 1 + (-4.83e9 + 2.79e9i)T^{2} \)
97 \( 1 + (8.54e4 + 2.29e4i)T + (7.43e9 + 4.29e9i)T^{2} \)
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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

−14.64925026228505203143967013845, −13.84029511888420161946234751223, −12.44508951856654935359265539440, −10.92587852683318683285770360216, −10.03468676279492694748770109662, −8.605224763742801688675638206454, −7.00709086706743639641817654342, −4.71819623257437878016919190860, −3.98407805014194352552265625554, −0.21987689046216300827951537943, 2.04531839059953127688606930646, 5.11523989274642329997540600750, 6.00677401973335996776964090322, 8.073253931645176916833077898866, 9.134727358667850419043488107619, 10.84990945397648662674519019002, 12.26614060596648848746921411889, 12.92813716251239603549115515573, 14.50236044302137135227601162819, 15.25668801881536925955942846128

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