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Properties

Label 2-140-5.4-c1-0-1
Degree 22
Conductor 140140
Sign 0.894−0.447i0.894 - 0.447i
Analytic cond. 1.117901.11790
Root an. cond. 1.057311.05731
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  + (1 + 2i)5-s + i·7-s + 3·9-s − 4i·13-s + 4i·17-s − 4·19-s − 8i·23-s + (−3 + 4i)25-s − 2·29-s − 8·31-s + (−2 + i)35-s − 8i·37-s + 6·41-s − 8i·43-s + (3 + 6i)45-s + ⋯
L(s)  = 1  + (0.447 + 0.894i)5-s + 0.377i·7-s + 9-s − 1.10i·13-s + 0.970i·17-s − 0.917·19-s − 1.66i·23-s + (−0.600 + 0.800i)25-s − 0.371·29-s − 1.43·31-s + (−0.338 + 0.169i)35-s − 1.31i·37-s + 0.937·41-s − 1.21i·43-s + (0.447 + 0.894i)45-s + ⋯

Functional equation

Λ(s)=(140s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(140s/2 ΓC(s+1/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 140140    =    22⋅5⋅72^{2} \cdot 5 \cdot 7
Sign: 0.894−0.447i0.894 - 0.447i
Analytic conductor: 1.117901.11790
Root analytic conductor: 1.057311.05731
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ140(29,⋅)\chi_{140} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 140, ( :1/2), 0.894−0.447i)(2,\ 140,\ (\ :1/2),\ 0.894 - 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.15984+0.273802i1.15984 + 0.273802i
L(12)L(\frac12) ≈\approx 1.15984+0.273802i1.15984 + 0.273802i
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
5 1+(−1−2i)T 1 + (-1 - 2i)T
7 1−iT 1 - iT
good3 1−3T2 1 - 3T^{2}
11 1+11T2 1 + 11T^{2}
13 1+4iT−13T2 1 + 4iT - 13T^{2}
17 1−4iT−17T2 1 - 4iT - 17T^{2}
19 1+4T+19T2 1 + 4T + 19T^{2}
23 1+8iT−23T2 1 + 8iT - 23T^{2}
29 1+2T+29T2 1 + 2T + 29T^{2}
31 1+8T+31T2 1 + 8T + 31T^{2}
37 1+8iT−37T2 1 + 8iT - 37T^{2}
41 1−6T+41T2 1 - 6T + 41T^{2}
43 1+8iT−43T2 1 + 8iT - 43T^{2}
47 1−8iT−47T2 1 - 8iT - 47T^{2}
53 1−53T2 1 - 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1+6T+61T2 1 + 6T + 61T^{2}
67 1−8iT−67T2 1 - 8iT - 67T^{2}
71 1−12T+71T2 1 - 12T + 71T^{2}
73 1−4iT−73T2 1 - 4iT - 73T^{2}
79 1−4T+79T2 1 - 4T + 79T^{2}
83 1−83T2 1 - 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1+12iT−97T2 1 + 12iT - 97T^{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

−12.96669995256079151075404019677, −12.56486098484899281556676128666, −10.79324829091812215006432572837, −10.45585803066577920268658596337, −9.172468262081889166146317305118, −7.85347736771192485577263380643, −6.70175068557014595121269523456, −5.65226771146563682921522801148, −3.93490860985937793236879916016, −2.27423480707383389006180586212, 1.67830909819770971279465816738, 4.03744434739196994906888648472, 5.12471343564457321137671406627, 6.62707383899723269380668387610, 7.73105409135580991251498714852, 9.187506760102842660251056287206, 9.732895421012609469365159122906, 11.10031189249695439529331808755, 12.18572119518809602416269541520, 13.19936138384739914382352658115

Graph of the ZZ-function along the critical line