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Properties

Label 2-156-1.1-c1-0-0
Degree 22
Conductor 156156
Sign 11
Analytic cond. 1.245661.24566
Root an. cond. 1.116091.11609
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 + 2·7-s + 9-s + 13-s − 6·17-s + 2·19-s + 2·21-s − 5·25-s + 27-s − 6·29-s + 2·31-s + 2·37-s + 39-s − 12·41-s − 4·43-s − 3·49-s − 6·51-s + 6·53-s + 2·57-s + 12·59-s + 2·61-s + 2·63-s − 10·67-s + 12·71-s + 14·73-s − 5·75-s + 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.277·13-s − 1.45·17-s + 0.458·19-s + 0.436·21-s − 25-s + 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.328·37-s + 0.160·39-s − 1.87·41-s − 0.609·43-s − 3/7·49-s − 0.840·51-s + 0.824·53-s + 0.264·57-s + 1.56·59-s + 0.256·61-s + 0.251·63-s − 1.22·67-s + 1.42·71-s + 1.63·73-s − 0.577·75-s + 0.900·79-s + ⋯

Functional equation

Λ(s)=(156s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(156s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 156156    =    22⋅3⋅132^{2} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 1.245661.24566
Root analytic conductor: 1.116091.11609
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 156, ( :1/2), 1)(2,\ 156,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.3761495931.376149593
L(12)L(\frac12) ≈\approx 1.3761495931.376149593
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
3 1−T 1 - T
13 1−T 1 - T
good5 1+pT2 1 + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+pT2 1 + 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+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1−2T+pT2 1 - 2 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+12T+pT2 1 + 12 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+10T+pT2 1 + 10 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

−13.21941002116462755470673769512, −11.83258307695703192104822918116, −11.02368824085056898890395801496, −9.798663464890123710839719924879, −8.735814680045936069504206621070, −7.86871640882441783507560416572, −6.65843891177652373785895186115, −5.11454570698520102977754794044, −3.80067854107620357152276821608, −2.03881758362108754018941589683, 2.03881758362108754018941589683, 3.80067854107620357152276821608, 5.11454570698520102977754794044, 6.65843891177652373785895186115, 7.86871640882441783507560416572, 8.735814680045936069504206621070, 9.798663464890123710839719924879, 11.02368824085056898890395801496, 11.83258307695703192104822918116, 13.21941002116462755470673769512

Graph of the ZZ-function along the critical line