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Properties

Label 2-58800-1.1-c1-0-122
Degree 22
Conductor 5880058800
Sign 11
Analytic cond. 469.520469.520
Root an. cond. 21.668421.6684
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 + 9-s + 5·11-s + 13-s − 2·17-s + 7·19-s + 3·23-s − 27-s − 6·31-s − 5·33-s + 5·37-s − 39-s + 9·41-s + 10·43-s + 13·47-s + 2·51-s + 53-s − 7·57-s + 4·59-s + 2·61-s + 6·67-s − 3·69-s + 2·71-s + 4·73-s + 14·79-s + 81-s + 10·83-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.50·11-s + 0.277·13-s − 0.485·17-s + 1.60·19-s + 0.625·23-s − 0.192·27-s − 1.07·31-s − 0.870·33-s + 0.821·37-s − 0.160·39-s + 1.40·41-s + 1.52·43-s + 1.89·47-s + 0.280·51-s + 0.137·53-s − 0.927·57-s + 0.520·59-s + 0.256·61-s + 0.733·67-s − 0.361·69-s + 0.237·71-s + 0.468·73-s + 1.57·79-s + 1/9·81-s + 1.09·83-s + ⋯

Functional equation

Λ(s)=(58800s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(58800s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5880058800    =    24⋅3⋅52⋅722^{4} \cdot 3 \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 469.520469.520
Root analytic conductor: 21.668421.6684
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 58800, ( :1/2), 1)(2,\ 58800,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0779192033.077919203
L(12)L(\frac12) ≈\approx 3.0779192033.077919203
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
5 1 1
7 1 1
good11 1−5T+pT2 1 - 5 T + p T^{2}
13 1−T+pT2 1 - T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1−7T+pT2 1 - 7 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+6T+pT2 1 + 6 T + p T^{2}
37 1−5T+pT2 1 - 5 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1−13T+pT2 1 - 13 T + p T^{2}
53 1−T+pT2 1 - T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1−6T+pT2 1 - 6 T + p T^{2}
71 1−2T+pT2 1 - 2 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1−14T+pT2 1 - 14 T + p T^{2}
83 1−10T+pT2 1 - 10 T + p T^{2}
89 1+10T+pT2 1 + 10 T + p T^{2}
97 1−8T+pT2 1 - 8 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

−14.26332438636786, −13.91809607330338, −13.33943298000872, −12.66144457670162, −12.32196691369626, −11.74532158570147, −11.24846199047695, −10.96700270274880, −10.34973272802419, −9.474915322654702, −9.260752858238102, −8.946629534318281, −7.936623231258914, −7.505377024349614, −6.897785094235104, −6.505836739811693, −5.702126892921779, −5.511108424719917, −4.633692694786946, −4.012802590584426, −3.666111001106252, −2.756051354102955, −2.028432497860164, −1.016954171216186, −0.8159488504394748, 0.8159488504394748, 1.016954171216186, 2.028432497860164, 2.756051354102955, 3.666111001106252, 4.012802590584426, 4.633692694786946, 5.511108424719917, 5.702126892921779, 6.505836739811693, 6.897785094235104, 7.505377024349614, 7.936623231258914, 8.946629534318281, 9.260752858238102, 9.474915322654702, 10.34973272802419, 10.96700270274880, 11.24846199047695, 11.74532158570147, 12.32196691369626, 12.66144457670162, 13.33943298000872, 13.91809607330338, 14.26332438636786

Graph of the ZZ-function along the critical line