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Properties

Label 2-39-13.4-c3-0-4
Degree $2$
Conductor $39$
Sign $0.842 - 0.538i$
Analytic cond. $2.30107$
Root an. cond. $1.51692$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.57 + 2.06i)2-s + (1.5 − 2.59i)3-s + (4.50 + 7.79i)4-s + 3.05i·5-s + (10.7 − 6.18i)6-s + (−5.78 + 3.34i)7-s + 4.12i·8-s + (−4.5 − 7.79i)9-s + (−6.28 + 10.8i)10-s + (−27.9 − 16.1i)11-s + 27.0·12-s + (−22.1 + 41.3i)13-s − 27.5·14-s + (7.92 + 4.57i)15-s + (27.4 − 47.6i)16-s + (−14.4 − 24.9i)17-s + ⋯
L(s)  = 1  + (1.26 + 0.728i)2-s + (0.288 − 0.499i)3-s + (0.562 + 0.974i)4-s + 0.272i·5-s + (0.728 − 0.420i)6-s + (−0.312 + 0.180i)7-s + 0.182i·8-s + (−0.166 − 0.288i)9-s + (−0.198 + 0.344i)10-s + (−0.765 − 0.441i)11-s + 0.649·12-s + (−0.472 + 0.881i)13-s − 0.526·14-s + (0.136 + 0.0787i)15-s + (0.429 − 0.744i)16-s + (−0.205 − 0.356i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(39\)    =    \(3 \cdot 13\)
Sign: $0.842 - 0.538i$
Analytic conductor: \(2.30107\)
Root analytic conductor: \(1.51692\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{39} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 39,\ (\ :3/2),\ 0.842 - 0.538i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.23466 + 0.652462i\)
\(L(\frac12)\) \(\approx\) \(2.23466 + 0.652462i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 2.59i)T \)
13 \( 1 + (22.1 - 41.3i)T \)
good2 \( 1 + (-3.57 - 2.06i)T + (4 + 6.92i)T^{2} \)
5 \( 1 - 3.05iT - 125T^{2} \)
7 \( 1 + (5.78 - 3.34i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (27.9 + 16.1i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (14.4 + 24.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-87.6 + 50.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (59.4 - 103. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (80.0 - 138. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 38.0iT - 2.97e4T^{2} \)
37 \( 1 + (-283. - 163. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-48.5 - 28.0i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-63.9 - 110. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 517. iT - 1.03e5T^{2} \)
53 \( 1 + 695.T + 1.48e5T^{2} \)
59 \( 1 + (-568. + 328. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (350. + 607. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (49.5 + 28.5i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (267. - 154. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 389. iT - 3.89e5T^{2} \)
79 \( 1 - 901.T + 4.93e5T^{2} \)
83 \( 1 + 687. iT - 5.71e5T^{2} \)
89 \( 1 + (927. + 535. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.51e3 - 877. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−15.62526117404837519491671176704, −14.47455915233071643916205065923, −13.65502986646947209428845529494, −12.74454881657731490412515082260, −11.47574450689179592881531640645, −9.476045523362915230513697705406, −7.60187918493724442768531567253, −6.52590347559654692379567273262, −5.07111828248195752304953152354, −3.11429660786754407067721342564, 2.76113370361538354469163619614, 4.32761933821869543892968411129, 5.62503951100183094920402058110, 7.955262631434725645731715206078, 9.834682235292705628733824368901, 10.90278905675412221686278518675, 12.38309530084941604137348254434, 13.11245334490740739837074166361, 14.32680304512919641170116703348, 15.25637736073738160766511101110

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