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Properties

Label 2-6300-1.1-c1-0-20
Degree 22
Conductor 63006300
Sign 11
Analytic cond. 50.305750.3057
Root an. cond. 7.092657.09265
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  − 7-s + 4·11-s + 6·13-s + 2·17-s + 6·19-s + 2·23-s − 6·29-s − 2·31-s + 4·37-s − 8·41-s + 4·43-s + 4·47-s + 49-s + 6·53-s − 4·59-s + 14·61-s − 4·67-s + 10·73-s − 4·77-s − 16·83-s − 8·89-s − 6·91-s − 10·97-s − 16·101-s − 16·103-s − 6·107-s − 14·109-s + ⋯
L(s)  = 1  − 0.377·7-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 1.37·19-s + 0.417·23-s − 1.11·29-s − 0.359·31-s + 0.657·37-s − 1.24·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.824·53-s − 0.520·59-s + 1.79·61-s − 0.488·67-s + 1.17·73-s − 0.455·77-s − 1.75·83-s − 0.847·89-s − 0.628·91-s − 1.01·97-s − 1.59·101-s − 1.57·103-s − 0.580·107-s − 1.34·109-s + ⋯

Functional equation

Λ(s)=(6300s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6300s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 63006300    =    22⋅32⋅52⋅72^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 50.305750.3057
Root analytic conductor: 7.092657.09265
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6300, ( :1/2), 1)(2,\ 6300,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.5071908932.507190893
L(12)L(\frac12) ≈\approx 2.5071908932.507190893
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 1
5 1 1
7 1+T 1 + T
good11 1−4T+pT2 1 - 4 T + p T^{2}
13 1−6T+pT2 1 - 6 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1−2T+pT2 1 - 2 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+2T+pT2 1 + 2 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1+8T+pT2 1 + 8 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−10T+pT2 1 - 10 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+16T+pT2 1 + 16 T + p T^{2}
89 1+8T+pT2 1 + 8 T + 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

−8.131890706453948184570671350528, −7.16210467109688053377700596946, −6.73729654778588428291582758126, −5.78786571439662084834813964485, −5.46505875227051947526810291933, −4.14934242766918376469827701972, −3.68069679041887771932599163856, −2.96269561686820682512049306286, −1.60945577397969897700417016725, −0.909971821433674117280248571734, 0.909971821433674117280248571734, 1.60945577397969897700417016725, 2.96269561686820682512049306286, 3.68069679041887771932599163856, 4.14934242766918376469827701972, 5.46505875227051947526810291933, 5.78786571439662084834813964485, 6.73729654778588428291582758126, 7.16210467109688053377700596946, 8.131890706453948184570671350528

Graph of the ZZ-function along the critical line