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Properties

Label 2-1098-9.7-c1-0-1
Degree 22
Conductor 10981098
Sign 0.144−0.989i0.144 - 0.989i
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  + (−0.5 + 0.866i)2-s + (−0.887 − 1.48i)3-s + (−0.499 − 0.866i)4-s + (−2.04 − 3.55i)5-s + (1.73 − 0.0253i)6-s + (−1.01 + 1.76i)7-s + 0.999·8-s + (−1.42 + 2.64i)9-s + 4.09·10-s + (−0.373 + 0.646i)11-s + (−0.843 + 1.51i)12-s + (−1.26 − 2.19i)13-s + (−1.01 − 1.76i)14-s + (−3.45 + 6.20i)15-s + (−0.5 + 0.866i)16-s − 6.94·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.512 − 0.858i)3-s + (−0.249 − 0.433i)4-s + (−0.916 − 1.58i)5-s + (0.707 − 0.0103i)6-s + (−0.384 + 0.666i)7-s + 0.353·8-s + (−0.474 + 0.880i)9-s + 1.29·10-s + (−0.112 + 0.194i)11-s + (−0.243 + 0.436i)12-s + (−0.351 − 0.609i)13-s + (−0.272 − 0.471i)14-s + (−0.893 + 1.60i)15-s + (−0.125 + 0.216i)16-s − 1.68·17-s + ⋯

Functional equation

Λ(s)=(1098s/2 ΓC(s) L(s)=((0.144−0.989i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1098s/2 ΓC(s+1/2) L(s)=((0.144−0.989i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10981098    =    2⋅32⋅612 \cdot 3^{2} \cdot 61
Sign: 0.144−0.989i0.144 - 0.989i
Analytic conductor: 8.767578.76757
Root analytic conductor: 2.961002.96100
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1098(367,⋅)\chi_{1098} (367, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1098, ( :1/2), 0.144−0.989i)(2,\ 1098,\ (\ :1/2),\ 0.144 - 0.989i)

Particular Values

L(1)L(1) ≈\approx 0.21320784060.2132078406
L(12)L(\frac12) ≈\approx 0.21320784060.2132078406
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
3 1+(0.887+1.48i)T 1 + (0.887 + 1.48i)T
61 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good5 1+(2.04+3.55i)T+(−2.5+4.33i)T2 1 + (2.04 + 3.55i)T + (-2.5 + 4.33i)T^{2}
7 1+(1.01−1.76i)T+(−3.5−6.06i)T2 1 + (1.01 - 1.76i)T + (-3.5 - 6.06i)T^{2}
11 1+(0.373−0.646i)T+(−5.5−9.52i)T2 1 + (0.373 - 0.646i)T + (-5.5 - 9.52i)T^{2}
13 1+(1.26+2.19i)T+(−6.5+11.2i)T2 1 + (1.26 + 2.19i)T + (-6.5 + 11.2i)T^{2}
17 1+6.94T+17T2 1 + 6.94T + 17T^{2}
19 1−4.45T+19T2 1 - 4.45T + 19T^{2}
23 1+(2.94+5.10i)T+(−11.5+19.9i)T2 1 + (2.94 + 5.10i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.87+3.25i)T+(−14.5−25.1i)T2 1 + (-1.87 + 3.25i)T + (-14.5 - 25.1i)T^{2}
31 1+(1.13+1.96i)T+(−15.5+26.8i)T2 1 + (1.13 + 1.96i)T + (-15.5 + 26.8i)T^{2}
37 1−0.890T+37T2 1 - 0.890T + 37T^{2}
41 1+(−4.15−7.19i)T+(−20.5+35.5i)T2 1 + (-4.15 - 7.19i)T + (-20.5 + 35.5i)T^{2}
43 1+(3.79−6.56i)T+(−21.5−37.2i)T2 1 + (3.79 - 6.56i)T + (-21.5 - 37.2i)T^{2}
47 1+(0.632−1.09i)T+(−23.5−40.7i)T2 1 + (0.632 - 1.09i)T + (-23.5 - 40.7i)T^{2}
53 1−1.98T+53T2 1 - 1.98T + 53T^{2}
59 1+(−4.87−8.44i)T+(−29.5+51.0i)T2 1 + (-4.87 - 8.44i)T + (-29.5 + 51.0i)T^{2}
67 1+(−5.67−9.82i)T+(−33.5+58.0i)T2 1 + (-5.67 - 9.82i)T + (-33.5 + 58.0i)T^{2}
71 1−6.97T+71T2 1 - 6.97T + 71T^{2}
73 1−9.19T+73T2 1 - 9.19T + 73T^{2}
79 1+(−3.16+5.47i)T+(−39.5−68.4i)T2 1 + (-3.16 + 5.47i)T + (-39.5 - 68.4i)T^{2}
83 1+(−8.05+13.9i)T+(−41.5−71.8i)T2 1 + (-8.05 + 13.9i)T + (-41.5 - 71.8i)T^{2}
89 1+2.90T+89T2 1 + 2.90T + 89T^{2}
97 1+(6.30−10.9i)T+(−48.5−84.0i)T2 1 + (6.30 - 10.9i)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

−9.741455557703674044662304782591, −8.961548620508075672288469939093, −8.208604946726224127667574676683, −7.77862401180558241803470225571, −6.73415903618349832685863930831, −5.85658108841534857598299595672, −5.02515065312083164348597914862, −4.33014106869550830249580196368, −2.44883805809955476028574788617, −0.929744907315696037268935977356, 0.14790336062001609750858090504, 2.46464805613640135614683297253, 3.66200261073216991152791623068, 3.86993964559070792945056842138, 5.17992597919712886098625372444, 6.66863161203667712358435152949, 6.97945006889652125687623728487, 8.040494023970602127086737057807, 9.180996449959244886758177444950, 9.889443694136875898378855464453

Graph of the ZZ-function along the critical line