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Properties

Label 2-1098-183.101-c1-0-2
Degree 22
Conductor 10981098
Sign −0.772−0.634i-0.772 - 0.634i
Analytic cond. 8.767578.76757
Root an. cond. 2.961002.96100
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (1.69 − 2.93i)5-s + (0.229 + 0.857i)7-s + (−0.707 + 0.707i)8-s + (−0.878 + 3.27i)10-s + (−4.13 + 4.13i)11-s + (−1.60 + 2.77i)13-s + (−0.443 − 0.768i)14-s + (0.500 − 0.866i)16-s + (−2.21 − 0.592i)17-s + (−6.05 + 3.49i)19-s − 3.39i·20-s + (2.92 − 5.07i)22-s + (−6.28 − 6.28i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (0.758 − 1.31i)5-s + (0.0868 + 0.324i)7-s + (−0.249 + 0.249i)8-s + (−0.277 + 1.03i)10-s + (−1.24 + 1.24i)11-s + (−0.444 + 0.769i)13-s + (−0.118 − 0.205i)14-s + (0.125 − 0.216i)16-s + (−0.536 − 0.143i)17-s + (−1.38 + 0.802i)19-s − 0.758i·20-s + (0.624 − 1.08i)22-s + (−1.31 − 1.31i)23-s + ⋯

Functional equation

Λ(s)=(1098s/2 ΓC(s) L(s)=((−0.772−0.634i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1098s/2 ΓC(s+1/2) L(s)=((−0.772−0.634i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10981098    =    2⋅32⋅612 \cdot 3^{2} \cdot 61
Sign: −0.772−0.634i-0.772 - 0.634i
Analytic conductor: 8.767578.76757
Root analytic conductor: 2.961002.96100
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1098(467,⋅)\chi_{1098} (467, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1098, ( :1/2), −0.772−0.634i)(2,\ 1098,\ (\ :1/2),\ -0.772 - 0.634i)

Particular Values

L(1)L(1) ≈\approx 0.33753870550.3375387055
L(12)L(\frac12) ≈\approx 0.33753870550.3375387055
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+(0.965−0.258i)T 1 + (0.965 - 0.258i)T
3 1 1
61 1+(6.52+4.29i)T 1 + (6.52 + 4.29i)T
good5 1+(−1.69+2.93i)T+(−2.5−4.33i)T2 1 + (-1.69 + 2.93i)T + (-2.5 - 4.33i)T^{2}
7 1+(−0.229−0.857i)T+(−6.06+3.5i)T2 1 + (-0.229 - 0.857i)T + (-6.06 + 3.5i)T^{2}
11 1+(4.13−4.13i)T−11iT2 1 + (4.13 - 4.13i)T - 11iT^{2}
13 1+(1.60−2.77i)T+(−6.5−11.2i)T2 1 + (1.60 - 2.77i)T + (-6.5 - 11.2i)T^{2}
17 1+(2.21+0.592i)T+(14.7+8.5i)T2 1 + (2.21 + 0.592i)T + (14.7 + 8.5i)T^{2}
19 1+(6.05−3.49i)T+(9.5−16.4i)T2 1 + (6.05 - 3.49i)T + (9.5 - 16.4i)T^{2}
23 1+(6.28+6.28i)T+23iT2 1 + (6.28 + 6.28i)T + 23iT^{2}
29 1+(−0.737−0.197i)T+(25.1+14.5i)T2 1 + (-0.737 - 0.197i)T + (25.1 + 14.5i)T^{2}
31 1+(2.79−10.4i)T+(−26.8−15.5i)T2 1 + (2.79 - 10.4i)T + (-26.8 - 15.5i)T^{2}
37 1+(−5.70−5.70i)T+37iT2 1 + (-5.70 - 5.70i)T + 37iT^{2}
41 1−0.414T+41T2 1 - 0.414T + 41T^{2}
43 1+(5.78−1.55i)T+(37.2−21.5i)T2 1 + (5.78 - 1.55i)T + (37.2 - 21.5i)T^{2}
47 1+(−2.33+1.34i)T+(23.5−40.7i)T2 1 + (-2.33 + 1.34i)T + (23.5 - 40.7i)T^{2}
53 1+(−3.97−3.97i)T+53iT2 1 + (-3.97 - 3.97i)T + 53iT^{2}
59 1+(1.53+5.72i)T+(−51.0+29.5i)T2 1 + (1.53 + 5.72i)T + (-51.0 + 29.5i)T^{2}
67 1+(−3.62+0.970i)T+(58.0−33.5i)T2 1 + (-3.62 + 0.970i)T + (58.0 - 33.5i)T^{2}
71 1+(2.13+0.572i)T+(61.4+35.5i)T2 1 + (2.13 + 0.572i)T + (61.4 + 35.5i)T^{2}
73 1+(0.946+1.63i)T+(−36.5+63.2i)T2 1 + (0.946 + 1.63i)T + (-36.5 + 63.2i)T^{2}
79 1+(3.48+13.0i)T+(−68.4+39.5i)T2 1 + (3.48 + 13.0i)T + (-68.4 + 39.5i)T^{2}
83 1+(3.05+1.76i)T+(41.5+71.8i)T2 1 + (3.05 + 1.76i)T + (41.5 + 71.8i)T^{2}
89 1+(10.7−10.7i)T−89iT2 1 + (10.7 - 10.7i)T - 89iT^{2}
97 1+(−12.7+7.38i)T+(48.5−84.0i)T2 1 + (-12.7 + 7.38i)T + (48.5 - 84.0i)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

−10.15529475914979778310564087641, −9.263838767695691920064556254463, −8.583075348689479189098126587734, −7.994991718582389989133376868743, −6.85399981513952028159352934124, −6.02052749693479341308457769664, −4.96011050431043910696321574117, −4.47024308533655477428824363575, −2.33577935029037892934094221257, −1.75997687535072148324420741627, 0.16595184278699914971705789770, 2.23364463579423133668569622820, 2.77966845114271660128937068336, 3.98481231853472073459092071796, 5.67294036744016437223936651250, 6.09556982616745824605575995515, 7.23696883892103829508723781813, 7.82894965041484630300192075849, 8.716194553185643126069778687336, 9.816527386345704693438536221327

Graph of the ZZ-function along the critical line