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Properties

Label 2-77-1.1-c1-0-1
Degree 22
Conductor 7777
Sign 11
Analytic cond. 0.6148480.614848
Root an. cond. 0.7841220.784122
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 3·5-s + 7-s − 2·9-s − 11-s − 2·12-s − 4·13-s + 3·15-s + 4·16-s − 6·17-s + 2·19-s − 6·20-s + 21-s + 3·23-s + 4·25-s − 5·27-s − 2·28-s − 6·29-s + 5·31-s − 33-s + 3·35-s + 4·36-s + 11·37-s − 4·39-s + 6·41-s + 8·43-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 1.34·5-s + 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.577·12-s − 1.10·13-s + 0.774·15-s + 16-s − 1.45·17-s + 0.458·19-s − 1.34·20-s + 0.218·21-s + 0.625·23-s + 4/5·25-s − 0.962·27-s − 0.377·28-s − 1.11·29-s + 0.898·31-s − 0.174·33-s + 0.507·35-s + 2/3·36-s + 1.80·37-s − 0.640·39-s + 0.937·41-s + 1.21·43-s + ⋯

Functional equation

Λ(s)=(77s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(77s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7777    =    7⋅117 \cdot 11
Sign: 11
Analytic conductor: 0.6148480.614848
Root analytic conductor: 0.7841220.784122
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 77, ( :1/2), 1)(2,\ 77,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0326067101.032606710
L(12)L(\frac12) ≈\approx 1.0326067101.032606710
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)
bad7 1−T 1 - T
11 1+T 1 + T
good2 1+pT2 1 + p T^{2}
3 1−T+pT2 1 - T + p T^{2}
5 1−3T+pT2 1 - 3 T + p T^{2}
13 1+4T+pT2 1 + 4 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−5T+pT2 1 - 5 T + p T^{2}
37 1−11T+pT2 1 - 11 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1−8T+pT2 1 - 8 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+9T+pT2 1 + 9 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1−5T+pT2 1 - 5 T + p T^{2}
71 1−9T+pT2 1 - 9 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+3T+pT2 1 + 3 T + p T^{2}
97 1+T+pT2 1 + T + p 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

−14.24843777324878748577425819366, −13.60450574831141336956421481732, −12.75123359214433885650815349898, −11.03928612573383528826645274943, −9.594904136117515823873522897188, −9.121939609810073781754575230776, −7.79543673470654027074280114997, −5.89237731397243587720148634695, −4.69940224529243356978622440045, −2.52489627324288051568453078479, 2.52489627324288051568453078479, 4.69940224529243356978622440045, 5.89237731397243587720148634695, 7.79543673470654027074280114997, 9.121939609810073781754575230776, 9.594904136117515823873522897188, 11.03928612573383528826645274943, 12.75123359214433885650815349898, 13.60450574831141336956421481732, 14.24843777324878748577425819366

Graph of the ZZ-function along the critical line