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Properties

Label 2-39-13.7-c2-0-4
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(7,⋅)\chi_{39} (7, \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

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

Graph of the ZZ-function along the critical line