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Properties

Label 2-39-13.2-c2-0-2
Degree 22
Conductor 3939
Sign 0.919−0.394i0.919 - 0.394i
Analytic cond. 1.062671.06267
Root an. cond. 1.030861.03086
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.72 + 0.463i)2-s + (0.866 + 1.5i)3-s + (−0.691 − 0.399i)4-s + (0.707 − 0.707i)5-s + (0.802 + 2.99i)6-s + (−2.02 + 0.543i)7-s + (−6.07 − 6.07i)8-s + (−1.5 + 2.59i)9-s + (1.55 − 0.895i)10-s + (2.74 − 10.2i)11-s − 1.38i·12-s + (−1.20 + 12.9i)13-s − 3.75·14-s + (1.67 + 0.448i)15-s + (−6.08 − 10.5i)16-s + (−8.98 − 5.18i)17-s + ⋯
L(s)  = 1  + (0.864 + 0.231i)2-s + (0.288 + 0.5i)3-s + (−0.172 − 0.0998i)4-s + (0.141 − 0.141i)5-s + (0.133 + 0.498i)6-s + (−0.289 + 0.0775i)7-s + (−0.758 − 0.758i)8-s + (−0.166 + 0.288i)9-s + (0.155 − 0.0895i)10-s + (0.249 − 0.931i)11-s − 0.115i·12-s + (−0.0925 + 0.995i)13-s − 0.268·14-s + (0.111 + 0.0299i)15-s + (−0.380 − 0.658i)16-s + (−0.528 − 0.304i)17-s + ⋯

Functional equation

Λ(s)=(39s/2 ΓC(s) L(s)=((0.919−0.394i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.394i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(39s/2 ΓC(s+1) L(s)=((0.919−0.394i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.919 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3939    =    3⋅133 \cdot 13
Sign: 0.919−0.394i0.919 - 0.394i
Analytic conductor: 1.062671.06267
Root analytic conductor: 1.030861.03086
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ39(28,⋅)\chi_{39} (28, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 39, ( :1), 0.919−0.394i)(2,\ 39,\ (\ :1),\ 0.919 - 0.394i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.45493+0.298846i1.45493 + 0.298846i
L(12)L(\frac12) ≈\approx 1.45493+0.298846i1.45493 + 0.298846i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−0.866−1.5i)T 1 + (-0.866 - 1.5i)T
13 1+(1.20−12.9i)T 1 + (1.20 - 12.9i)T
good2 1+(−1.72−0.463i)T+(3.46+2i)T2 1 + (-1.72 - 0.463i)T + (3.46 + 2i)T^{2}
5 1+(−0.707+0.707i)T−25iT2 1 + (-0.707 + 0.707i)T - 25iT^{2}
7 1+(2.02−0.543i)T+(42.4−24.5i)T2 1 + (2.02 - 0.543i)T + (42.4 - 24.5i)T^{2}
11 1+(−2.74+10.2i)T+(−104.−60.5i)T2 1 + (-2.74 + 10.2i)T + (-104. - 60.5i)T^{2}
17 1+(8.98+5.18i)T+(144.5+250.i)T2 1 + (8.98 + 5.18i)T + (144.5 + 250. i)T^{2}
19 1+(−5.44−20.3i)T+(−312.+180.5i)T2 1 + (-5.44 - 20.3i)T + (-312. + 180.5i)T^{2}
23 1+(−21.6+12.4i)T+(264.5−458.i)T2 1 + (-21.6 + 12.4i)T + (264.5 - 458. i)T^{2}
29 1+(−12.1−20.9i)T+(−420.5+728.i)T2 1 + (-12.1 - 20.9i)T + (-420.5 + 728. i)T^{2}
31 1+(−18.1+18.1i)T−961iT2 1 + (-18.1 + 18.1i)T - 961iT^{2}
37 1+(−7.82+29.2i)T+(−1.18e3−684.5i)T2 1 + (-7.82 + 29.2i)T + (-1.18e3 - 684.5i)T^{2}
41 1+(−23.9−6.43i)T+(1.45e3+840.5i)T2 1 + (-23.9 - 6.43i)T + (1.45e3 + 840.5i)T^{2}
43 1+(72.3+41.7i)T+(924.5+1.60e3i)T2 1 + (72.3 + 41.7i)T + (924.5 + 1.60e3i)T^{2}
47 1+(45.3+45.3i)T+2.20e3iT2 1 + (45.3 + 45.3i)T + 2.20e3iT^{2}
53 1+65.1T+2.80e3T2 1 + 65.1T + 2.80e3T^{2}
59 1+(−74.5+19.9i)T+(3.01e3−1.74e3i)T2 1 + (-74.5 + 19.9i)T + (3.01e3 - 1.74e3i)T^{2}
61 1+(−13.2+23.0i)T+(−1.86e3−3.22e3i)T2 1 + (-13.2 + 23.0i)T + (-1.86e3 - 3.22e3i)T^{2}
67 1+(−74.9−20.0i)T+(3.88e3+2.24e3i)T2 1 + (-74.9 - 20.0i)T + (3.88e3 + 2.24e3i)T^{2}
71 1+(27.4+102.i)T+(−4.36e3+2.52e3i)T2 1 + (27.4 + 102. i)T + (-4.36e3 + 2.52e3i)T^{2}
73 1+(−82.5−82.5i)T+5.32e3iT2 1 + (-82.5 - 82.5i)T + 5.32e3iT^{2}
79 1+70.2T+6.24e3T2 1 + 70.2T + 6.24e3T^{2}
83 1+(−9.18+9.18i)T−6.88e3iT2 1 + (-9.18 + 9.18i)T - 6.88e3iT^{2}
89 1+(14.7−55.1i)T+(−6.85e3−3.96e3i)T2 1 + (14.7 - 55.1i)T + (-6.85e3 - 3.96e3i)T^{2}
97 1+(11.7+43.6i)T+(−8.14e3+4.70e3i)T2 1 + (11.7 + 43.6i)T + (-8.14e3 + 4.70e3i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−15.91353982482922048932140289316, −14.68613438070839078904209658002, −13.91711122540962545897873420322, −12.88355439851058989835281786426, −11.41112849165503445209337238060, −9.743355319017968805068222537288, −8.739533123004451261399699513109, −6.52827343986116804744454988478, −5.07246054828626169489668773777, −3.56769911569006616679307730878, 2.92169342248754909725123137733, 4.79039634929267519553093428192, 6.58690827256320507364615294741, 8.246624445899892142262824226560, 9.726751381064605614379747358780, 11.50741728523114003888227966359, 12.74746057620785631238627450389, 13.34625695598510133387796402438, 14.56176035026512926694462219145, 15.49775564830306010078881840343

Graph of the ZZ-function along the critical line