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Properties

Label 2-2e7-8.5-c3-0-5
Degree 22
Conductor 128128
Sign 11
Analytic cond. 7.552247.55224
Root an. cond. 2.748132.74813
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.32i·3-s − 17.8i·5-s + 22.6·7-s − 13.0·9-s − 44.2i·11-s + 17.8i·13-s + 113.·15-s + 70·17-s + 82.2i·19-s + 143. i·21-s + 158.·23-s − 195.·25-s + 88.5i·27-s − 125. i·29-s + 280·33-s + ⋯
L(s)  = 1  + 1.21i·3-s − 1.59i·5-s + 1.22·7-s − 0.481·9-s − 1.21i·11-s + 0.381i·13-s + 1.94·15-s + 0.998·17-s + 0.992i·19-s + 1.48i·21-s + 1.43·23-s − 1.56·25-s + 0.631i·27-s − 0.801i·29-s + 1.47·33-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=( Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+3/2) L(s)=( Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: 11
Analytic conductor: 7.552247.55224
Root analytic conductor: 2.748132.74813
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ128(65,⋅)\chi_{128} (65, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 128, ( :3/2), 1)(2,\ 128,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.867331.86733
L(12)L(\frac12) ≈\approx 1.867331.86733
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 1
good3 1−6.32iT−27T2 1 - 6.32iT - 27T^{2}
5 1+17.8iT−125T2 1 + 17.8iT - 125T^{2}
7 1−22.6T+343T2 1 - 22.6T + 343T^{2}
11 1+44.2iT−1.33e3T2 1 + 44.2iT - 1.33e3T^{2}
13 1−17.8iT−2.19e3T2 1 - 17.8iT - 2.19e3T^{2}
17 1−70T+4.91e3T2 1 - 70T + 4.91e3T^{2}
19 1−82.2iT−6.85e3T2 1 - 82.2iT - 6.85e3T^{2}
23 1−158.T+1.21e4T2 1 - 158.T + 1.21e4T^{2}
29 1+125.iT−2.43e4T2 1 + 125. iT - 2.43e4T^{2}
31 1+2.97e4T2 1 + 2.97e4T^{2}
37 1+375.iT−5.06e4T2 1 + 375. iT - 5.06e4T^{2}
41 1+182T+6.89e4T2 1 + 182T + 6.89e4T^{2}
43 1−132.iT−7.95e4T2 1 - 132. iT - 7.95e4T^{2}
47 1+316.T+1.03e5T2 1 + 316.T + 1.03e5T^{2}
53 1−125.iT−1.48e5T2 1 - 125. iT - 1.48e5T^{2}
59 1−82.2iT−2.05e5T2 1 - 82.2iT - 2.05e5T^{2}
61 1−232.iT−2.26e5T2 1 - 232. iT - 2.26e5T^{2}
67 1+221.iT−3.00e5T2 1 + 221. iT - 3.00e5T^{2}
71 1−113.T+3.57e5T2 1 - 113.T + 3.57e5T^{2}
73 1−910T+3.89e5T2 1 - 910T + 3.89e5T^{2}
79 1+678.T+4.93e5T2 1 + 678.T + 4.93e5T^{2}
83 1−714.iT−5.71e5T2 1 - 714. iT - 5.71e5T^{2}
89 1+546T+7.04e5T2 1 + 546T + 7.04e5T^{2}
97 1+490T+9.12e5T2 1 + 490T + 9.12e5T^{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

−12.81146615570764980042204002957, −11.70129097841085557184618340537, −10.82447452772290597494812629115, −9.582775670188518468128407765530, −8.730562439661298931194605571401, −7.934803100557612782905712947515, −5.57790411167408496988752528083, −4.87596700180645501866596888215, −3.79679415022851530481763655912, −1.20462889427626375237706648641, 1.58060051165689216341964464967, 2.91788851574558275541648851809, 5.00896723109477689334745653313, 6.74195622579686793005980169405, 7.22580022671621646331627904796, 8.163832432087228483075589236271, 9.949161790519663999598469526772, 10.99007053273707497041302588095, 11.80680442221122422453218854430, 12.87451666649714569753020263937

Graph of the ZZ-function along the critical line