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Properties

Label 2-70-1.1-c3-0-2
Degree 22
Conductor 7070
Sign 11
Analytic cond. 4.130134.13013
Root an. cond. 2.032272.03227
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 5·3-s + 4·4-s + 5·5-s + 10·6-s − 7·7-s + 8·8-s − 2·9-s + 10·10-s − 11-s + 20·12-s + 7·13-s − 14·14-s + 25·15-s + 16·16-s − 51·17-s − 4·18-s + 30·19-s + 20·20-s − 35·21-s − 2·22-s − 50·23-s + 40·24-s + 25·25-s + 14·26-s − 145·27-s − 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.962·3-s + 1/2·4-s + 0.447·5-s + 0.680·6-s − 0.377·7-s + 0.353·8-s − 0.0740·9-s + 0.316·10-s − 0.0274·11-s + 0.481·12-s + 0.149·13-s − 0.267·14-s + 0.430·15-s + 1/4·16-s − 0.727·17-s − 0.0523·18-s + 0.362·19-s + 0.223·20-s − 0.363·21-s − 0.0193·22-s − 0.453·23-s + 0.340·24-s + 1/5·25-s + 0.105·26-s − 1.03·27-s − 0.188·28-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 4.130134.13013
Root analytic conductor: 2.032272.03227
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 70, ( :3/2), 1)(2,\ 70,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.7629723712.762972371
L(12)L(\frac12) ≈\approx 2.7629723712.762972371
L(52)L(\frac{5}{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−pT 1 - p T
5 1−pT 1 - p T
7 1+pT 1 + p T
good3 1−5T+p3T2 1 - 5 T + p^{3} T^{2}
11 1+T+p3T2 1 + T + p^{3} T^{2}
13 1−7T+p3T2 1 - 7 T + p^{3} T^{2}
17 1+3pT+p3T2 1 + 3 p T + p^{3} T^{2}
19 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
23 1+50T+p3T2 1 + 50 T + p^{3} T^{2}
29 1−79T+p3T2 1 - 79 T + p^{3} T^{2}
31 1+212T+p3T2 1 + 212 T + p^{3} T^{2}
37 1+190T+p3T2 1 + 190 T + p^{3} T^{2}
41 1+308T+p3T2 1 + 308 T + p^{3} T^{2}
43 1−422T+p3T2 1 - 422 T + p^{3} T^{2}
47 1−121T+p3T2 1 - 121 T + p^{3} T^{2}
53 1−664T+p3T2 1 - 664 T + p^{3} T^{2}
59 1−628T+p3T2 1 - 628 T + p^{3} T^{2}
61 1+684T+p3T2 1 + 684 T + p^{3} T^{2}
67 1−1056T+p3T2 1 - 1056 T + p^{3} T^{2}
71 1−744T+p3T2 1 - 744 T + p^{3} T^{2}
73 1−726T+p3T2 1 - 726 T + p^{3} T^{2}
79 1+407T+p3T2 1 + 407 T + p^{3} T^{2}
83 1−644T+p3T2 1 - 644 T + p^{3} T^{2}
89 1+880T+p3T2 1 + 880 T + p^{3} T^{2}
97 1+1351T+p3T2 1 + 1351 T + p^{3} 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

−14.02877161294499100316632927655, −13.44594765697618113745660725958, −12.29312693211253801967469387058, −10.91024816717137091636211064660, −9.554797129906410198586835726130, −8.434841000984463701430404146740, −6.95881400190180065163549875993, −5.53867578225588174848636045224, −3.73453652326893131111598181444, −2.33720109746932864701269806985, 2.33720109746932864701269806985, 3.73453652326893131111598181444, 5.53867578225588174848636045224, 6.95881400190180065163549875993, 8.434841000984463701430404146740, 9.554797129906410198586835726130, 10.91024816717137091636211064660, 12.29312693211253801967469387058, 13.44594765697618113745660725958, 14.02877161294499100316632927655

Graph of the ZZ-function along the critical line