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Properties

Label 2-300-1.1-c3-0-8
Degree 22
Conductor 300300
Sign −1-1
Analytic cond. 17.700517.7005
Root an. cond. 4.207204.20720
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 7·7-s + 9·9-s − 54·11-s − 55·13-s + 18·17-s − 25·19-s − 21·21-s − 18·23-s + 27·27-s − 54·29-s − 271·31-s − 162·33-s + 314·37-s − 165·39-s − 360·41-s − 163·43-s + 522·47-s − 294·49-s + 54·51-s − 36·53-s − 75·57-s + 126·59-s + 47·61-s − 63·63-s − 343·67-s − 54·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.48·11-s − 1.17·13-s + 0.256·17-s − 0.301·19-s − 0.218·21-s − 0.163·23-s + 0.192·27-s − 0.345·29-s − 1.57·31-s − 0.854·33-s + 1.39·37-s − 0.677·39-s − 1.37·41-s − 0.578·43-s + 1.62·47-s − 6/7·49-s + 0.148·51-s − 0.0933·53-s − 0.174·57-s + 0.278·59-s + 0.0986·61-s − 0.125·63-s − 0.625·67-s − 0.0942·69-s + ⋯

Functional equation

Λ(s)=(300s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(300s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 300300    =    22⋅3⋅522^{2} \cdot 3 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 17.700517.7005
Root analytic conductor: 4.207204.20720
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 300, ( :3/2), −1)(2,\ 300,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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−pT 1 - p T
5 1 1
good7 1+pT+p3T2 1 + p T + p^{3} T^{2}
11 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
13 1+55T+p3T2 1 + 55 T + p^{3} T^{2}
17 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
19 1+25T+p3T2 1 + 25 T + p^{3} T^{2}
23 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
29 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
31 1+271T+p3T2 1 + 271 T + p^{3} T^{2}
37 1−314T+p3T2 1 - 314 T + p^{3} T^{2}
41 1+360T+p3T2 1 + 360 T + p^{3} T^{2}
43 1+163T+p3T2 1 + 163 T + p^{3} T^{2}
47 1−522T+p3T2 1 - 522 T + p^{3} T^{2}
53 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
59 1−126T+p3T2 1 - 126 T + p^{3} T^{2}
61 1−47T+p3T2 1 - 47 T + p^{3} T^{2}
67 1+343T+p3T2 1 + 343 T + p^{3} T^{2}
71 1+1080T+p3T2 1 + 1080 T + p^{3} T^{2}
73 1+1054T+p3T2 1 + 1054 T + p^{3} T^{2}
79 1+568T+p3T2 1 + 568 T + p^{3} T^{2}
83 1−1422T+p3T2 1 - 1422 T + p^{3} T^{2}
89 1−1440T+p3T2 1 - 1440 T + p^{3} T^{2}
97 1+439T+p3T2 1 + 439 T + p^{3} 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

−10.60393524382082800308740318364, −9.933335741707288806558710574607, −8.973556349541966096752627848731, −7.84215172273293491935428795961, −7.20901275437343674262649268955, −5.74768931925189855809602543867, −4.67087257949738426258614610369, −3.21995671807754163112183796659, −2.16935379911079178249702886515, 0, 2.16935379911079178249702886515, 3.21995671807754163112183796659, 4.67087257949738426258614610369, 5.74768931925189855809602543867, 7.20901275437343674262649268955, 7.84215172273293491935428795961, 8.973556349541966096752627848731, 9.933335741707288806558710574607, 10.60393524382082800308740318364

Graph of the ZZ-function along the critical line