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Properties

Label 2-1100-1.1-c5-0-27
Degree 22
Conductor 11001100
Sign 11
Analytic cond. 176.422176.422
Root an. cond. 13.282413.2824
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 25.2·3-s − 200.·7-s + 395.·9-s − 121·11-s − 727.·13-s + 1.10e3·17-s + 1.66e3·19-s − 5.07e3·21-s − 2.98e3·23-s + 3.85e3·27-s − 1.55e3·29-s + 9.51e3·31-s − 3.05e3·33-s + 9.43e3·37-s − 1.83e4·39-s + 7.37e3·41-s + 8.52e3·43-s − 3.00e4·47-s + 2.35e4·49-s + 2.80e4·51-s − 2.39e4·53-s + 4.20e4·57-s − 6.96e3·59-s + 4.90e4·61-s − 7.94e4·63-s + 2.39e4·67-s − 7.54e4·69-s + ⋯
L(s)  = 1  + 1.62·3-s − 1.54·7-s + 1.62·9-s − 0.301·11-s − 1.19·13-s + 0.930·17-s + 1.05·19-s − 2.51·21-s − 1.17·23-s + 1.01·27-s − 0.342·29-s + 1.77·31-s − 0.488·33-s + 1.13·37-s − 1.93·39-s + 0.684·41-s + 0.703·43-s − 1.98·47-s + 1.39·49-s + 1.50·51-s − 1.17·53-s + 1.71·57-s − 0.260·59-s + 1.68·61-s − 2.52·63-s + 0.651·67-s − 1.90·69-s + ⋯

Functional equation

Λ(s)=(1100s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(1100s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11001100    =    22⋅52⋅112^{2} \cdot 5^{2} \cdot 11
Sign: 11
Analytic conductor: 176.422176.422
Root analytic conductor: 13.282413.2824
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1100, ( :5/2), 1)(2,\ 1100,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.2006002633.200600263
L(12)L(\frac12) ≈\approx 3.2006002633.200600263
L(72)L(\frac{7}{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
5 1 1
11 1+121T 1 + 121T
good3 1−25.2T+243T2 1 - 25.2T + 243T^{2}
7 1+200.T+1.68e4T2 1 + 200.T + 1.68e4T^{2}
13 1+727.T+3.71e5T2 1 + 727.T + 3.71e5T^{2}
17 1−1.10e3T+1.41e6T2 1 - 1.10e3T + 1.41e6T^{2}
19 1−1.66e3T+2.47e6T2 1 - 1.66e3T + 2.47e6T^{2}
23 1+2.98e3T+6.43e6T2 1 + 2.98e3T + 6.43e6T^{2}
29 1+1.55e3T+2.05e7T2 1 + 1.55e3T + 2.05e7T^{2}
31 1−9.51e3T+2.86e7T2 1 - 9.51e3T + 2.86e7T^{2}
37 1−9.43e3T+6.93e7T2 1 - 9.43e3T + 6.93e7T^{2}
41 1−7.37e3T+1.15e8T2 1 - 7.37e3T + 1.15e8T^{2}
43 1−8.52e3T+1.47e8T2 1 - 8.52e3T + 1.47e8T^{2}
47 1+3.00e4T+2.29e8T2 1 + 3.00e4T + 2.29e8T^{2}
53 1+2.39e4T+4.18e8T2 1 + 2.39e4T + 4.18e8T^{2}
59 1+6.96e3T+7.14e8T2 1 + 6.96e3T + 7.14e8T^{2}
61 1−4.90e4T+8.44e8T2 1 - 4.90e4T + 8.44e8T^{2}
67 1−2.39e4T+1.35e9T2 1 - 2.39e4T + 1.35e9T^{2}
71 1+1.88e3T+1.80e9T2 1 + 1.88e3T + 1.80e9T^{2}
73 1−1.36e4T+2.07e9T2 1 - 1.36e4T + 2.07e9T^{2}
79 1−1.15e4T+3.07e9T2 1 - 1.15e4T + 3.07e9T^{2}
83 1−5.51e4T+3.93e9T2 1 - 5.51e4T + 3.93e9T^{2}
89 1+2.37e3T+5.58e9T2 1 + 2.37e3T + 5.58e9T^{2}
97 1−7.91e3T+8.58e9T2 1 - 7.91e3T + 8.58e9T^{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

−9.426604749672404054954676420340, −8.143860377733046143408726517928, −7.72951357722036142332244718628, −6.83243892641194877071893411846, −5.85516457346152632802635669192, −4.57456401131460419077485394512, −3.46693229221865785686066318833, −2.96367061050953404735041271258, −2.17194814833003514639695452126, −0.67825551052578481813377483905, 0.67825551052578481813377483905, 2.17194814833003514639695452126, 2.96367061050953404735041271258, 3.46693229221865785686066318833, 4.57456401131460419077485394512, 5.85516457346152632802635669192, 6.83243892641194877071893411846, 7.72951357722036142332244718628, 8.143860377733046143408726517928, 9.426604749672404054954676420340

Graph of the ZZ-function along the critical line