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Properties

Label 2-150-1.1-c7-0-11
Degree 22
Conductor 150150
Sign −1-1
Analytic cond. 46.857746.8577
Root an. cond. 6.845276.84527
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 27·3-s + 64·4-s + 216·6-s − 349·7-s − 512·8-s + 729·9-s + 1.18e3·11-s − 1.72e3·12-s − 1.72e3·13-s + 2.79e3·14-s + 4.09e3·16-s − 7.49e3·17-s − 5.83e3·18-s + 1.27e4·19-s + 9.42e3·21-s − 9.45e3·22-s + 6.40e3·23-s + 1.38e4·24-s + 1.37e4·26-s − 1.96e4·27-s − 2.23e4·28-s + 1.08e5·29-s + 1.42e5·31-s − 3.27e4·32-s − 3.19e4·33-s + 5.99e4·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.384·7-s − 0.353·8-s + 1/3·9-s + 0.267·11-s − 0.288·12-s − 0.217·13-s + 0.271·14-s + 1/4·16-s − 0.369·17-s − 0.235·18-s + 0.427·19-s + 0.222·21-s − 0.189·22-s + 0.109·23-s + 0.204·24-s + 0.153·26-s − 0.192·27-s − 0.192·28-s + 0.822·29-s + 0.858·31-s − 0.176·32-s − 0.154·33-s + 0.261·34-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: −1-1
Analytic conductor: 46.857746.8577
Root analytic conductor: 6.845276.84527
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 150, ( :7/2), −1)(2,\ 150,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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+p3T 1 + p^{3} T
3 1+p3T 1 + p^{3} T
5 1 1
good7 1+349T+p7T2 1 + 349 T + p^{7} T^{2}
11 1−1182T+p7T2 1 - 1182 T + p^{7} T^{2}
13 1+1723T+p7T2 1 + 1723 T + p^{7} T^{2}
17 1+7494T+p7T2 1 + 7494 T + p^{7} T^{2}
19 1−12785T+p7T2 1 - 12785 T + p^{7} T^{2}
23 1−6402T+p7T2 1 - 6402 T + p^{7} T^{2}
29 1−108090T+p7T2 1 - 108090 T + p^{7} T^{2}
31 1−142427T+p7T2 1 - 142427 T + p^{7} T^{2}
37 1−276266T+p7T2 1 - 276266 T + p^{7} T^{2}
41 1−525072T+p7T2 1 - 525072 T + p^{7} T^{2}
43 1+747013T+p7T2 1 + 747013 T + p^{7} T^{2}
47 1−571326T+p7T2 1 - 571326 T + p^{7} T^{2}
53 1+1472028T+p7T2 1 + 1472028 T + p^{7} T^{2}
59 1+1582110T+p7T2 1 + 1582110 T + p^{7} T^{2}
61 1+932893T+p7T2 1 + 932893 T + p^{7} T^{2}
67 1+1688089T+p7T2 1 + 1688089 T + p^{7} T^{2}
71 1−2962752T+p7T2 1 - 2962752 T + p^{7} T^{2}
73 1+4078798T+p7T2 1 + 4078798 T + p^{7} T^{2}
79 1+5635360T+p7T2 1 + 5635360 T + p^{7} T^{2}
83 1+3120318T+p7T2 1 + 3120318 T + p^{7} T^{2}
89 1+9155040T+p7T2 1 + 9155040 T + p^{7} T^{2}
97 1+10041199T+p7T2 1 + 10041199 T + p^{7} 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

−11.12694845589798813694123368819, −10.09880566160050835078353020577, −9.276093177671324661450215698427, −8.043075379459003587699181091881, −6.88886532025233144731988666985, −5.98949402032303233485521144938, −4.54052090392755121003541752833, −2.88641641666643973884806475781, −1.27871572286614365546431515389, 0, 1.27871572286614365546431515389, 2.88641641666643973884806475781, 4.54052090392755121003541752833, 5.98949402032303233485521144938, 6.88886532025233144731988666985, 8.043075379459003587699181091881, 9.276093177671324661450215698427, 10.09880566160050835078353020577, 11.12694845589798813694123368819

Graph of the ZZ-function along the critical line