↑

Properties

Label 2-39-13.9-c1-0-1
Degree 22
Conductor 3939
Sign 0.859−0.511i0.859 - 0.511i
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  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (0.500 + 0.866i)4-s − 5-s + (0.499 + 0.866i)6-s + (−1 − 1.73i)7-s − 3·8-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)10-s + (1 − 1.73i)11-s + 12-s + (−3.5 − 0.866i)13-s + 1.99·14-s + (−0.5 + 0.866i)15-s + (0.500 − 0.866i)16-s + (3.5 + 6.06i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (0.250 + 0.433i)4-s − 0.447·5-s + (0.204 + 0.353i)6-s + (−0.377 − 0.654i)7-s − 1.06·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.301 − 0.522i)11-s + 0.288·12-s + (−0.970 − 0.240i)13-s + 0.534·14-s + (−0.129 + 0.223i)15-s + (0.125 − 0.216i)16-s + (0.848 + 1.47i)17-s + ⋯

Functional equation

Λ(s)=(39s/2 ΓC(s) L(s)=((0.859−0.511i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(39s/2 ΓC(s+1/2) L(s)=((0.859−0.511i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 3939    =    3⋅133 \cdot 13
Sign: 0.859−0.511i0.859 - 0.511i
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), 0.859−0.511i)(2,\ 39,\ (\ :1/2),\ 0.859 - 0.511i)

Particular Values

L(1)L(1) ≈\approx 0.658958+0.181103i0.658958 + 0.181103i
L(12)L(\frac12) ≈\approx 0.658958+0.181103i0.658958 + 0.181103i
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+(3.5+0.866i)T 1 + (3.5 + 0.866i)T
good2 1+(0.5−0.866i)T+(−1−1.73i)T2 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2}
5 1+T+5T2 1 + T + 5T^{2}
7 1+(1+1.73i)T+(−3.5+6.06i)T2 1 + (1 + 1.73i)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+(−3.5−6.06i)T+(−8.5+14.7i)T2 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2}
19 1+(−3−5.19i)T+(−9.5+16.4i)T2 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2}
23 1+(−3+5.19i)T+(−11.5−19.9i)T2 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2}
29 1+(−0.5+0.866i)T+(−14.5−25.1i)T2 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+(0.5−0.866i)T+(−18.5−32.0i)T2 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2}
41 1+(4.5−7.79i)T+(−20.5−35.5i)T2 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2}
43 1+(3+5.19i)T+(−21.5+37.2i)T2 1 + (3 + 5.19i)T + (-21.5 + 37.2i)T^{2}
47 1−6T+47T2 1 - 6T + 47T^{2}
53 1+9T+53T2 1 + 9T + 53T^{2}
59 1+(−29.5+51.0i)T2 1 + (-29.5 + 51.0i)T^{2}
61 1+(0.5+0.866i)T+(−30.5+52.8i)T2 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1+1.73i)T+(−33.5−58.0i)T2 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2}
71 1+(3+5.19i)T+(−35.5+61.4i)T2 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2}
73 1−11T+73T2 1 - 11T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1+14T+83T2 1 + 14T + 83T^{2}
89 1+(−7+12.1i)T+(−44.5−77.0i)T2 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2}
97 1+(−1−1.73i)T+(−48.5+84.0i)T2 1 + (-1 - 1.73i)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

−16.67964485929451701471305897548, −15.31436868569418472237139205121, −14.25352282411028397136124411118, −12.73289175444341950296811916307, −11.86794614489910264294294396928, −10.08439462454078301203589831316, −8.363264865171006164651980684627, −7.54477102164470802446718447887, −6.24892583756641342435496526807, −3.47426894868114625087598578954, 2.86393153909535063400258284725, 5.20432204657102172725166254858, 7.22047791579711791825933304934, 9.254398779241532270121720047097, 9.791768698484293397611172234114, 11.42867064720421448825118723410, 12.14357937143744666454386491571, 14.02499101548283092459780281457, 15.27243809376413943797361286045, 15.84408544761433403349579987483

Graph of the ZZ-function along the critical line