↑

Properties

Label 2-980-140.107-c1-0-22
Degree 22
Conductor 980980
Sign −0.451−0.892i-0.451 - 0.892i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
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  + (1.02 − 0.970i)2-s + (−0.517 + 1.93i)3-s + (0.115 − 1.99i)4-s + (0.653 + 2.13i)5-s + (1.34 + 2.49i)6-s + (−1.81 − 2.16i)8-s + (−0.870 − 0.502i)9-s + (2.74 + 1.56i)10-s + (−4.92 + 2.84i)11-s + (3.79 + 1.25i)12-s + (−4.82 + 4.82i)13-s + (−4.47 + 0.155i)15-s + (−3.97 − 0.463i)16-s + (0.979 − 3.65i)17-s + (−1.38 + 0.327i)18-s + (0.564 − 0.977i)19-s + ⋯
L(s)  = 1  + (0.727 − 0.686i)2-s + (−0.299 + 1.11i)3-s + (0.0579 − 0.998i)4-s + (0.292 + 0.956i)5-s + (0.548 + 1.01i)6-s + (−0.642 − 0.765i)8-s + (−0.290 − 0.167i)9-s + (0.868 + 0.494i)10-s + (−1.48 + 0.856i)11-s + (1.09 + 0.363i)12-s + (−1.33 + 1.33i)13-s + (−1.15 + 0.0401i)15-s + (−0.993 − 0.115i)16-s + (0.237 − 0.886i)17-s + (−0.325 + 0.0772i)18-s + (0.129 − 0.224i)19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.451−0.892i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.451−0.892i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.451−0.892i-0.451 - 0.892i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(667,⋅)\chi_{980} (667, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.451−0.892i)(2,\ 980,\ (\ :1/2),\ -0.451 - 0.892i)

Particular Values

L(1)L(1) ≈\approx 0.699074+1.13656i0.699074 + 1.13656i
L(12)L(\frac12) ≈\approx 0.699074+1.13656i0.699074 + 1.13656i
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)
bad2 1+(−1.02+0.970i)T 1 + (-1.02 + 0.970i)T
5 1+(−0.653−2.13i)T 1 + (-0.653 - 2.13i)T
7 1 1
good3 1+(0.517−1.93i)T+(−2.59−1.5i)T2 1 + (0.517 - 1.93i)T + (-2.59 - 1.5i)T^{2}
11 1+(4.92−2.84i)T+(5.5−9.52i)T2 1 + (4.92 - 2.84i)T + (5.5 - 9.52i)T^{2}
13 1+(4.82−4.82i)T−13iT2 1 + (4.82 - 4.82i)T - 13iT^{2}
17 1+(−0.979+3.65i)T+(−14.7−8.5i)T2 1 + (-0.979 + 3.65i)T + (-14.7 - 8.5i)T^{2}
19 1+(−0.564+0.977i)T+(−9.5−16.4i)T2 1 + (-0.564 + 0.977i)T + (-9.5 - 16.4i)T^{2}
23 1+(−4.70+1.26i)T+(19.9−11.5i)T2 1 + (-4.70 + 1.26i)T + (19.9 - 11.5i)T^{2}
29 1+0.781iT−29T2 1 + 0.781iT - 29T^{2}
31 1+(3.20−1.84i)T+(15.5−26.8i)T2 1 + (3.20 - 1.84i)T + (15.5 - 26.8i)T^{2}
37 1+(−3.32+0.889i)T+(32.0−18.5i)T2 1 + (-3.32 + 0.889i)T + (32.0 - 18.5i)T^{2}
41 1+0.451T+41T2 1 + 0.451T + 41T^{2}
43 1+(0.613+0.613i)T+43iT2 1 + (0.613 + 0.613i)T + 43iT^{2}
47 1+(−1.10−4.14i)T+(−40.7+23.5i)T2 1 + (-1.10 - 4.14i)T + (-40.7 + 23.5i)T^{2}
53 1+(2.03+0.544i)T+(45.8+26.5i)T2 1 + (2.03 + 0.544i)T + (45.8 + 26.5i)T^{2}
59 1+(−4.63−8.03i)T+(−29.5+51.0i)T2 1 + (-4.63 - 8.03i)T + (-29.5 + 51.0i)T^{2}
61 1+(0.759−1.31i)T+(−30.5−52.8i)T2 1 + (0.759 - 1.31i)T + (-30.5 - 52.8i)T^{2}
67 1+(−13.1−3.53i)T+(58.0+33.5i)T2 1 + (-13.1 - 3.53i)T + (58.0 + 33.5i)T^{2}
71 1−11.8iT−71T2 1 - 11.8iT - 71T^{2}
73 1+(−1.66−0.445i)T+(63.2+36.5i)T2 1 + (-1.66 - 0.445i)T + (63.2 + 36.5i)T^{2}
79 1+(2.73−4.73i)T+(−39.5−68.4i)T2 1 + (2.73 - 4.73i)T + (-39.5 - 68.4i)T^{2}
83 1+(−0.968−0.968i)T+83iT2 1 + (-0.968 - 0.968i)T + 83iT^{2}
89 1+(8.16+4.71i)T+(44.5+77.0i)T2 1 + (8.16 + 4.71i)T + (44.5 + 77.0i)T^{2}
97 1+(−6.77−6.77i)T+97iT2 1 + (-6.77 - 6.77i)T + 97iT^{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

−10.25387685715272608189901762919, −9.805430550011990423639494490262, −9.247168342334730183511200268334, −7.36448868481133367545655835759, −6.89566536911466071959304946941, −5.46882872695057910866104165262, −4.93193517616907740616632529657, −4.19016869671033955357273309699, −2.89517231148849223461142107518, −2.21473286189472305932663795410, 0.46175636440420256575565203673, 2.20614063930517380206871810357, 3.34528596857595692507204736814, 4.89923595170796239125643146299, 5.46585111803510875893323161647, 6.09814695171933795800734810291, 7.27948189833807458709974706349, 7.912460005093465188053374078166, 8.356992996863981840515146387757, 9.628884123669378019884156454768

Graph of the ZZ-function along the critical line