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Properties

Label 2-39-13.6-c2-0-4
Degree 22
Conductor 3939
Sign −0.640+0.767i-0.640 + 0.767i
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  + (−0.816 − 3.04i)2-s + (0.866 − 1.5i)3-s + (−5.15 + 2.97i)4-s + (1.44 − 1.44i)5-s + (−5.27 − 1.41i)6-s + (−1.58 + 5.91i)7-s + (4.36 + 4.36i)8-s + (−1.5 − 2.59i)9-s + (−5.59 − 3.23i)10-s + (16.5 − 4.42i)11-s + 10.3i·12-s + (12.0 − 4.94i)13-s + 19.3·14-s + (−0.918 − 3.42i)15-s + (−2.16 + 3.75i)16-s + (−24.4 + 14.1i)17-s + ⋯
L(s)  = 1  + (−0.408 − 1.52i)2-s + (0.288 − 0.5i)3-s + (−1.28 + 0.744i)4-s + (0.289 − 0.289i)5-s + (−0.879 − 0.235i)6-s + (−0.226 + 0.844i)7-s + (0.546 + 0.546i)8-s + (−0.166 − 0.288i)9-s + (−0.559 − 0.323i)10-s + (1.50 − 0.402i)11-s + 0.859i·12-s + (0.924 − 0.380i)13-s + 1.38·14-s + (−0.0612 − 0.228i)15-s + (−0.135 + 0.234i)16-s + (−1.43 + 0.829i)17-s + ⋯

Functional equation

Λ(s)=(39s/2 ΓC(s) L(s)=((−0.640+0.767i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(39s/2 ΓC(s+1) L(s)=((−0.640+0.767i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3939    =    3⋅133 \cdot 13
Sign: −0.640+0.767i-0.640 + 0.767i
Analytic conductor: 1.062671.06267
Root analytic conductor: 1.030861.03086
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ39(19,⋅)\chi_{39} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 39, ( :1), −0.640+0.767i)(2,\ 39,\ (\ :1),\ -0.640 + 0.767i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.401221−0.857163i0.401221 - 0.857163i
L(12)L(\frac12) ≈\approx 0.401221−0.857163i0.401221 - 0.857163i
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+(−12.0+4.94i)T 1 + (-12.0 + 4.94i)T
good2 1+(0.816+3.04i)T+(−3.46+2i)T2 1 + (0.816 + 3.04i)T + (-3.46 + 2i)T^{2}
5 1+(−1.44+1.44i)T−25iT2 1 + (-1.44 + 1.44i)T - 25iT^{2}
7 1+(1.58−5.91i)T+(−42.4−24.5i)T2 1 + (1.58 - 5.91i)T + (-42.4 - 24.5i)T^{2}
11 1+(−16.5+4.42i)T+(104.−60.5i)T2 1 + (-16.5 + 4.42i)T + (104. - 60.5i)T^{2}
17 1+(24.4−14.1i)T+(144.5−250.i)T2 1 + (24.4 - 14.1i)T + (144.5 - 250. i)T^{2}
19 1+(−15.6−4.20i)T+(312.+180.5i)T2 1 + (-15.6 - 4.20i)T + (312. + 180.5i)T^{2}
23 1+(16.5+9.55i)T+(264.5+458.i)T2 1 + (16.5 + 9.55i)T + (264.5 + 458. i)T^{2}
29 1+(9.62−16.6i)T+(−420.5−728.i)T2 1 + (9.62 - 16.6i)T + (-420.5 - 728. i)T^{2}
31 1+(4.19−4.19i)T−961iT2 1 + (4.19 - 4.19i)T - 961iT^{2}
37 1+(40.5−10.8i)T+(1.18e3−684.5i)T2 1 + (40.5 - 10.8i)T + (1.18e3 - 684.5i)T^{2}
41 1+(2.32+8.67i)T+(−1.45e3+840.5i)T2 1 + (2.32 + 8.67i)T + (-1.45e3 + 840.5i)T^{2}
43 1+(−47.3+27.3i)T+(924.5−1.60e3i)T2 1 + (-47.3 + 27.3i)T + (924.5 - 1.60e3i)T^{2}
47 1+(10.9+10.9i)T+2.20e3iT2 1 + (10.9 + 10.9i)T + 2.20e3iT^{2}
53 1+58.0T+2.80e3T2 1 + 58.0T + 2.80e3T^{2}
59 1+(4.32−16.1i)T+(−3.01e3−1.74e3i)T2 1 + (4.32 - 16.1i)T + (-3.01e3 - 1.74e3i)T^{2}
61 1+(42.5+73.6i)T+(−1.86e3+3.22e3i)T2 1 + (42.5 + 73.6i)T + (-1.86e3 + 3.22e3i)T^{2}
67 1+(6.52+24.3i)T+(−3.88e3+2.24e3i)T2 1 + (6.52 + 24.3i)T + (-3.88e3 + 2.24e3i)T^{2}
71 1+(−67.6−18.1i)T+(4.36e3+2.52e3i)T2 1 + (-67.6 - 18.1i)T + (4.36e3 + 2.52e3i)T^{2}
73 1+(73.9+73.9i)T+5.32e3iT2 1 + (73.9 + 73.9i)T + 5.32e3iT^{2}
79 1−0.739T+6.24e3T2 1 - 0.739T + 6.24e3T^{2}
83 1+(−29.6+29.6i)T−6.88e3iT2 1 + (-29.6 + 29.6i)T - 6.88e3iT^{2}
89 1+(102.−27.5i)T+(6.85e3−3.96e3i)T2 1 + (102. - 27.5i)T + (6.85e3 - 3.96e3i)T^{2}
97 1+(−60.9−16.3i)T+(8.14e3+4.70e3i)T2 1 + (-60.9 - 16.3i)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

−15.56805615593219281907746695821, −13.93522212397359215027669916331, −12.84905685470703025873794808092, −11.95683116322309945405538383685, −10.88886369008489379252977106127, −9.233615554687916245643762006009, −8.685261434636637493061787180752, −6.23791029039975453731866571142, −3.57362183956764071032954922214, −1.67551174135104980136730032165, 4.26261708654315337460284569324, 6.26327211528674902318134667200, 7.24617950669371948751281824785, 8.861471773932817403790417644166, 9.723214511951638311461118942601, 11.40034834297343016450655418527, 13.75564102611775922121845351503, 14.20165080707734702612855782544, 15.55214034593229068143939822690, 16.28030321183000778028073430762

Graph of the ZZ-function along the critical line