Properties

Label 2-1098-183.101-c1-0-20
Degree 22
Conductor 10981098
Sign 0.123+0.992i-0.123 + 0.992i
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 + (2.02 − 3.51i)5-s + (−0.376 − 1.40i)7-s + (0.707 − 0.707i)8-s + (1.04 − 3.91i)10-s + (−2.07 + 2.07i)11-s + (2.10 − 3.64i)13-s + (−0.728 − 1.26i)14-s + (0.500 − 0.866i)16-s + (2.30 + 0.617i)17-s + (−3.11 + 1.80i)19-s − 4.05i·20-s + (−1.46 + 2.54i)22-s + (4.74 + 4.74i)23-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.906 − 1.57i)5-s + (−0.142 − 0.531i)7-s + (0.249 − 0.249i)8-s + (0.331 − 1.23i)10-s + (−0.626 + 0.626i)11-s + (0.584 − 1.01i)13-s + (−0.194 − 0.337i)14-s + (0.125 − 0.216i)16-s + (0.558 + 0.149i)17-s + (−0.715 + 0.413i)19-s − 0.906i·20-s + (−0.313 + 0.542i)22-s + (0.989 + 0.989i)23-s + ⋯

Functional equation

Λ(s)=(1098s/2ΓC(s)L(s)=((0.123+0.992i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1098s/2ΓC(s+1/2)L(s)=((0.123+0.992i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10981098    =    232612 \cdot 3^{2} \cdot 61
Sign: 0.123+0.992i-0.123 + 0.992i
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), 0.123+0.992i)(2,\ 1098,\ (\ :1/2),\ -0.123 + 0.992i)

Particular Values

L(1)L(1) \approx 2.7277889902.727788990
L(12)L(\frac12) \approx 2.7277889902.727788990
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)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+(7.312.73i)T 1 + (-7.31 - 2.73i)T
good5 1+(2.02+3.51i)T+(2.54.33i)T2 1 + (-2.02 + 3.51i)T + (-2.5 - 4.33i)T^{2}
7 1+(0.376+1.40i)T+(6.06+3.5i)T2 1 + (0.376 + 1.40i)T + (-6.06 + 3.5i)T^{2}
11 1+(2.072.07i)T11iT2 1 + (2.07 - 2.07i)T - 11iT^{2}
13 1+(2.10+3.64i)T+(6.511.2i)T2 1 + (-2.10 + 3.64i)T + (-6.5 - 11.2i)T^{2}
17 1+(2.300.617i)T+(14.7+8.5i)T2 1 + (-2.30 - 0.617i)T + (14.7 + 8.5i)T^{2}
19 1+(3.111.80i)T+(9.516.4i)T2 1 + (3.11 - 1.80i)T + (9.5 - 16.4i)T^{2}
23 1+(4.744.74i)T+23iT2 1 + (-4.74 - 4.74i)T + 23iT^{2}
29 1+(5.85+1.56i)T+(25.1+14.5i)T2 1 + (5.85 + 1.56i)T + (25.1 + 14.5i)T^{2}
31 1+(0.02330.0871i)T+(26.815.5i)T2 1 + (0.0233 - 0.0871i)T + (-26.8 - 15.5i)T^{2}
37 1+(2.072.07i)T+37iT2 1 + (-2.07 - 2.07i)T + 37iT^{2}
41 1+11.6T+41T2 1 + 11.6T + 41T^{2}
43 1+(4.34+1.16i)T+(37.221.5i)T2 1 + (-4.34 + 1.16i)T + (37.2 - 21.5i)T^{2}
47 1+(2.19+1.26i)T+(23.540.7i)T2 1 + (-2.19 + 1.26i)T + (23.5 - 40.7i)T^{2}
53 1+(1.83+1.83i)T+53iT2 1 + (1.83 + 1.83i)T + 53iT^{2}
59 1+(3.2712.2i)T+(51.0+29.5i)T2 1 + (-3.27 - 12.2i)T + (-51.0 + 29.5i)T^{2}
67 1+(5.71+1.53i)T+(58.033.5i)T2 1 + (-5.71 + 1.53i)T + (58.0 - 33.5i)T^{2}
71 1+(8.882.38i)T+(61.4+35.5i)T2 1 + (-8.88 - 2.38i)T + (61.4 + 35.5i)T^{2}
73 1+(0.863+1.49i)T+(36.5+63.2i)T2 1 + (0.863 + 1.49i)T + (-36.5 + 63.2i)T^{2}
79 1+(1.68+6.27i)T+(68.4+39.5i)T2 1 + (1.68 + 6.27i)T + (-68.4 + 39.5i)T^{2}
83 1+(3.26+1.88i)T+(41.5+71.8i)T2 1 + (3.26 + 1.88i)T + (41.5 + 71.8i)T^{2}
89 1+(13.313.3i)T89iT2 1 + (13.3 - 13.3i)T - 89iT^{2}
97 1+(9.18+5.30i)T+(48.584.0i)T2 1 + (-9.18 + 5.30i)T + (48.5 - 84.0i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.878026682265753194797233696563, −8.829849129226760141618132991572, −8.046421047471175574464115141916, −7.06888913332312251926688887309, −5.74174258261306644267879263219, −5.43974277938961408961019387471, −4.50456221051529759184907493737, −3.49752547946673453837440911268, −2.05404714295386899282687792114, −0.994193829111731350084876661334, 2.06909773723131838195310790798, 2.83747001715280827838800226784, 3.71408318154853550421706034203, 5.13426397045171488736088943837, 5.94984869498114793101197211555, 6.59472486565498946303863426785, 7.19466468984692916001912857251, 8.418973979691715045908159555552, 9.315548559315554857813782665001, 10.24621021929270968741249610776

Graph of the ZZ-function along the critical line