↑

Properties

Label 2-39-13.9-c1-0-0
Degree 22
Conductor 3939
Sign −0.597−0.802i-0.597 - 0.802i
Analytic cond. 0.3114160.311416
Root an. cond. 0.5580470.558047
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 + 2.21i)2-s + (−0.5 + 0.866i)3-s + (−2.28 − 3.95i)4-s + 0.561·5-s + (−1.28 − 2.21i)6-s + (1.78 + 3.08i)7-s + 6.56·8-s + (−0.499 − 0.866i)9-s + (−0.719 + 1.24i)10-s + (1 − 1.73i)11-s + 4.56·12-s + (0.5 − 3.57i)13-s − 9.12·14-s + (−0.280 + 0.486i)15-s + (−3.84 + 6.65i)16-s + (−1.28 − 2.21i)17-s + ⋯
L(s)  = 1  + (−0.905 + 1.56i)2-s + (−0.288 + 0.499i)3-s + (−1.14 − 1.97i)4-s + 0.251·5-s + (−0.522 − 0.905i)6-s + (0.673 + 1.16i)7-s + 2.31·8-s + (−0.166 − 0.288i)9-s + (−0.227 + 0.393i)10-s + (0.301 − 0.522i)11-s + 1.31·12-s + (0.138 − 0.990i)13-s − 2.43·14-s + (−0.0724 + 0.125i)15-s + (−0.960 + 1.66i)16-s + (−0.310 − 0.538i)17-s + ⋯

Functional equation

Λ(s)=(39s/2 ΓC(s) L(s)=((−0.597−0.802i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(39s/2 ΓC(s+1/2) L(s)=((−0.597−0.802i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3939    =    3⋅133 \cdot 13
Sign: −0.597−0.802i-0.597 - 0.802i
Analytic conductor: 0.3114160.311416
Root analytic conductor: 0.5580470.558047
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ39(22,⋅)\chi_{39} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 39, ( :1/2), −0.597−0.802i)(2,\ 39,\ (\ :1/2),\ -0.597 - 0.802i)

Particular Values

L(1)L(1) ≈\approx 0.221424+0.440843i0.221424 + 0.440843i
L(12)L(\frac12) ≈\approx 0.221424+0.440843i0.221424 + 0.440843i
L(32)L(\frac{3}{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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
13 1+(−0.5+3.57i)T 1 + (-0.5 + 3.57i)T
good2 1+(1.28−2.21i)T+(−1−1.73i)T2 1 + (1.28 - 2.21i)T + (-1 - 1.73i)T^{2}
5 1−0.561T+5T2 1 - 0.561T + 5T^{2}
7 1+(−1.78−3.08i)T+(−3.5+6.06i)T2 1 + (-1.78 - 3.08i)T + (-3.5 + 6.06i)T^{2}
11 1+(−1+1.73i)T+(−5.5−9.52i)T2 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2}
17 1+(1.28+2.21i)T+(−8.5+14.7i)T2 1 + (1.28 + 2.21i)T + (-8.5 + 14.7i)T^{2}
19 1+(−0.561−0.972i)T+(−9.5+16.4i)T2 1 + (-0.561 - 0.972i)T + (-9.5 + 16.4i)T^{2}
23 1+(1−1.73i)T+(−11.5−19.9i)T2 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.84+4.92i)T+(−14.5−25.1i)T2 1 + (-2.84 + 4.92i)T + (-14.5 - 25.1i)T^{2}
31 1+1.56T+31T2 1 + 1.56T + 31T^{2}
37 1+(1.71−2.97i)T+(−18.5−32.0i)T2 1 + (1.71 - 2.97i)T + (-18.5 - 32.0i)T^{2}
41 1+(1.28−2.21i)T+(−20.5−35.5i)T2 1 + (1.28 - 2.21i)T + (-20.5 - 35.5i)T^{2}
43 1+(0.219+0.379i)T+(−21.5+37.2i)T2 1 + (0.219 + 0.379i)T + (-21.5 + 37.2i)T^{2}
47 1+8.24T+47T2 1 + 8.24T + 47T^{2}
53 1−11.6T+53T2 1 - 11.6T + 53T^{2}
59 1+(−5.56−9.63i)T+(−29.5+51.0i)T2 1 + (-5.56 - 9.63i)T + (-29.5 + 51.0i)T^{2}
61 1+(6.06+10.4i)T+(−30.5+52.8i)T2 1 + (6.06 + 10.4i)T + (-30.5 + 52.8i)T^{2}
67 1+(0.219−0.379i)T+(−33.5−58.0i)T2 1 + (0.219 - 0.379i)T + (-33.5 - 58.0i)T^{2}
71 1+(7+12.1i)T+(−35.5+61.4i)T2 1 + (7 + 12.1i)T + (-35.5 + 61.4i)T^{2}
73 1+1.87T+73T2 1 + 1.87T + 73T^{2}
79 1−9.56T+79T2 1 - 9.56T + 79T^{2}
83 1+9.12T+83T2 1 + 9.12T + 83T^{2}
89 1+(6.56−11.3i)T+(−44.5−77.0i)T2 1 + (6.56 - 11.3i)T + (-44.5 - 77.0i)T^{2}
97 1+(−2.21−3.84i)T+(−48.5+84.0i)T2 1 + (-2.21 - 3.84i)T + (-48.5 + 84.0i)T^{2}
show more
show less
   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

−16.63626487814764417731471398613, −15.57813499304656079020648439903, −15.01285821919102920629488857213, −13.75745221868974112782545901432, −11.65894298933109523607952369996, −10.08321046175364581921012691368, −8.947228123916774784562183180588, −7.961425274891963000628013599450, −6.13080867782957238623680716028, −5.22682145542702382543323010512, 1.69954342443032167672892412906, 4.19046505752440223713785140131, 7.16476381016400549389812101067, 8.593978564601701248595039586842, 10.00983854422326925471949820373, 11.01306185846334421911876392666, 11.92839061625033255481999988837, 13.14467812921236505569820267523, 14.19216404935030640251005927216, 16.62083944234154749659699482497

Graph of the ZZ-function along the critical line