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Properties

Label 2-3960-440.109-c0-0-8
Degree 22
Conductor 39603960
Sign ii
Analytic cond. 1.976291.97629
Root an. cond. 1.405801.40580
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s − i·5-s + 2·7-s + i·8-s − 10-s + i·11-s − 2i·14-s + 16-s + i·20-s + 22-s − 25-s − 2·28-s + 2·31-s − i·32-s + ⋯
L(s)  = 1  − i·2-s − 4-s − i·5-s + 2·7-s + i·8-s − 10-s + i·11-s − 2i·14-s + 16-s + i·20-s + 22-s − 25-s − 2·28-s + 2·31-s − i·32-s + ⋯

Functional equation

Λ(s)=(3960s/2 ΓC(s) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3960s/2 ΓC(s) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 39603960    =    23⋅32⋅5⋅112^{3} \cdot 3^{2} \cdot 5 \cdot 11
Sign: ii
Analytic conductor: 1.976291.97629
Root analytic conductor: 1.405801.40580
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3960(109,⋅)\chi_{3960} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3960, ( :0), i)(2,\ 3960,\ (\ :0),\ i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5053366061.505336606
L(12)L(\frac12) ≈\approx 1.5053366061.505336606
L(1)L(1) 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+iT 1 + iT
3 1 1
5 1+iT 1 + iT
11 1−iT 1 - iT
good7 1−2T+T2 1 - 2T + T^{2}
13 1−T2 1 - T^{2}
17 1+T2 1 + T^{2}
19 1+T2 1 + T^{2}
23 1−T2 1 - T^{2}
29 1+T2 1 + T^{2}
31 1−2T+T2 1 - 2T + T^{2}
37 1+T2 1 + T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1−T2 1 - T^{2}
53 1+T2 1 + T^{2}
59 1−2iT−T2 1 - 2iT - T^{2}
61 1+T2 1 + T^{2}
67 1+T2 1 + T^{2}
71 1+T2 1 + T^{2}
73 1+2T+T2 1 + 2T + T^{2}
79 1−T2 1 - T^{2}
83 1+2iT−T2 1 + 2iT - T^{2}
89 1+T2 1 + T^{2}
97 1−T2 1 - 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

−8.580414721650511879668141091532, −7.961816500586970972920250543258, −7.36518257735511160934061330984, −5.87011552526503377948278442886, −5.09413490146369778547891403753, −4.50394828099472526858382166202, −4.21384490743126231566117374288, −2.67413679028139330027831027805, −1.76413640980303348561429978669, −1.15558673807892511001808746899, 1.19201641050947408877953704260, 2.50981962471622191764769163691, 3.61688609705468385581572613931, 4.48338386408034951493369863720, 5.17147362267924256959092263516, 5.93729789456770226597164268036, 6.62835101086678887862020391294, 7.43773208708229555394699808320, 8.164318686580281447868451982385, 8.332597739475181929107133254285

Graph of the ZZ-function along the critical line