↑

Properties

Label 2-3e4-81.13-c1-0-2
Degree 22
Conductor 8181
Sign 0.761−0.647i0.761 - 0.647i
Analytic cond. 0.6467880.646788
Root an. cond. 0.8042310.804231
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  + (0.744 + 0.373i)2-s + (0.703 + 1.58i)3-s + (−0.779 − 1.04i)4-s + (0.345 + 1.15i)5-s + (−0.0676 + 1.44i)6-s + (−0.520 − 1.20i)7-s + (−0.478 − 2.71i)8-s + (−2.00 + 2.22i)9-s + (−0.174 + 0.989i)10-s + (2.11 − 0.501i)11-s + (1.10 − 1.97i)12-s + (−3.80 − 2.50i)13-s + (0.0636 − 1.09i)14-s + (−1.58 + 1.36i)15-s + (−0.0912 + 0.304i)16-s + (3.54 − 1.28i)17-s + ⋯
L(s)  = 1  + (0.526 + 0.264i)2-s + (0.406 + 0.913i)3-s + (−0.389 − 0.523i)4-s + (0.154 + 0.516i)5-s + (−0.0276 + 0.588i)6-s + (−0.196 − 0.456i)7-s + (−0.169 − 0.958i)8-s + (−0.669 + 0.742i)9-s + (−0.0551 + 0.312i)10-s + (0.637 − 0.151i)11-s + (0.320 − 0.569i)12-s + (−1.05 − 0.694i)13-s + (0.0170 − 0.292i)14-s + (−0.409 + 0.351i)15-s + (−0.0228 + 0.0761i)16-s + (0.859 − 0.312i)17-s + ⋯

Functional equation

Λ(s)=(81s/2 ΓC(s) L(s)=((0.761−0.647i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(81s/2 ΓC(s+1/2) L(s)=((0.761−0.647i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8181    =    343^{4}
Sign: 0.761−0.647i0.761 - 0.647i
Analytic conductor: 0.6467880.646788
Root analytic conductor: 0.8042310.804231
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ81(13,⋅)\chi_{81} (13, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 81, ( :1/2), 0.761−0.647i)(2,\ 81,\ (\ :1/2),\ 0.761 - 0.647i)

Particular Values

L(1)L(1) ≈\approx 1.13740+0.418027i1.13740 + 0.418027i
L(12)L(\frac12) ≈\approx 1.13740+0.418027i1.13740 + 0.418027i
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+(−0.703−1.58i)T 1 + (-0.703 - 1.58i)T
good2 1+(−0.744−0.373i)T+(1.19+1.60i)T2 1 + (-0.744 - 0.373i)T + (1.19 + 1.60i)T^{2}
5 1+(−0.345−1.15i)T+(−4.17+2.74i)T2 1 + (-0.345 - 1.15i)T + (-4.17 + 2.74i)T^{2}
7 1+(0.520+1.20i)T+(−4.80+5.09i)T2 1 + (0.520 + 1.20i)T + (-4.80 + 5.09i)T^{2}
11 1+(−2.11+0.501i)T+(9.82−4.93i)T2 1 + (-2.11 + 0.501i)T + (9.82 - 4.93i)T^{2}
13 1+(3.80+2.50i)T+(5.14+11.9i)T2 1 + (3.80 + 2.50i)T + (5.14 + 11.9i)T^{2}
17 1+(−3.54+1.28i)T+(13.0−10.9i)T2 1 + (-3.54 + 1.28i)T + (13.0 - 10.9i)T^{2}
19 1+(2.50+0.911i)T+(14.5+12.2i)T2 1 + (2.50 + 0.911i)T + (14.5 + 12.2i)T^{2}
23 1+(2.38−5.53i)T+(−15.7−16.7i)T2 1 + (2.38 - 5.53i)T + (-15.7 - 16.7i)T^{2}
29 1+(−0.241−4.15i)T+(−28.8+3.36i)T2 1 + (-0.241 - 4.15i)T + (-28.8 + 3.36i)T^{2}
31 1+(7.40−0.865i)T+(30.1−7.14i)T2 1 + (7.40 - 0.865i)T + (30.1 - 7.14i)T^{2}
37 1+(−7.47−6.27i)T+(6.42+36.4i)T2 1 + (-7.47 - 6.27i)T + (6.42 + 36.4i)T^{2}
41 1+(−5.17+2.59i)T+(24.4−32.8i)T2 1 + (-5.17 + 2.59i)T + (24.4 - 32.8i)T^{2}
43 1+(−3.46−3.67i)T+(−2.50+42.9i)T2 1 + (-3.46 - 3.67i)T + (-2.50 + 42.9i)T^{2}
47 1+(−2.97−0.348i)T+(45.7+10.8i)T2 1 + (-2.97 - 0.348i)T + (45.7 + 10.8i)T^{2}
53 1+(6.22−10.7i)T+(−26.5−45.8i)T2 1 + (6.22 - 10.7i)T + (-26.5 - 45.8i)T^{2}
59 1+(10.6+2.51i)T+(52.7+26.4i)T2 1 + (10.6 + 2.51i)T + (52.7 + 26.4i)T^{2}
61 1+(−7.04+9.45i)T+(−17.4−58.4i)T2 1 + (-7.04 + 9.45i)T + (-17.4 - 58.4i)T^{2}
67 1+(−0.0482+0.828i)T+(−66.5−7.77i)T2 1 + (-0.0482 + 0.828i)T + (-66.5 - 7.77i)T^{2}
71 1+(−1.25+7.14i)T+(−66.7−24.2i)T2 1 + (-1.25 + 7.14i)T + (-66.7 - 24.2i)T^{2}
73 1+(1.41+7.99i)T+(−68.5+24.9i)T2 1 + (1.41 + 7.99i)T + (-68.5 + 24.9i)T^{2}
79 1+(10.2+5.16i)T+(47.1+63.3i)T2 1 + (10.2 + 5.16i)T + (47.1 + 63.3i)T^{2}
83 1+(4.26+2.14i)T+(49.5+66.5i)T2 1 + (4.26 + 2.14i)T + (49.5 + 66.5i)T^{2}
89 1+(−0.578−3.28i)T+(−83.6+30.4i)T2 1 + (-0.578 - 3.28i)T + (-83.6 + 30.4i)T^{2}
97 1+(0.390−1.30i)T+(−81.0−53.3i)T2 1 + (0.390 - 1.30i)T + (-81.0 - 53.3i)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

−14.49798531887954687776046986410, −13.88519213860512241837981232303, −12.59344630341229735809923326573, −10.91127604426353929450572479093, −9.996849513610254956064548183517, −9.225182394302638019747765128797, −7.48250397780019784360537381140, −5.91812524041052484590700227905, −4.67442667456820543059632374447, −3.31757211898587386784856436161, 2.39507635433655029372911158548, 4.16314192954395286162154840234, 5.82944469430959558126534426016, 7.39741712352485313953142164376, 8.600371352579214778816855464459, 9.449623837623398221116256944433, 11.55078706773173010556821584477, 12.59305741633429464623828691485, 12.73991577883053082202673608077, 14.28152636617441902296481287398

Graph of the ZZ-function along the critical line