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Properties

Label 2-150-1.1-c1-0-0
Degree 22
Conductor 150150
Sign 11
Analytic cond. 1.197751.19775
Root an. cond. 1.094421.09442
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  − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s + 2·11-s − 12-s + 6·13-s − 2·14-s + 16-s + 2·17-s − 18-s − 2·21-s − 2·22-s − 4·23-s + 24-s − 6·26-s − 27-s + 2·28-s − 8·31-s − 32-s − 2·33-s − 2·34-s + 36-s + 2·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s + 1.66·13-s − 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.436·21-s − 0.426·22-s − 0.834·23-s + 0.204·24-s − 1.17·26-s − 0.192·27-s + 0.377·28-s − 1.43·31-s − 0.176·32-s − 0.348·33-s − 0.342·34-s + 1/6·36-s + 0.328·37-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: 11
Analytic conductor: 1.197751.19775
Root analytic conductor: 1.094421.09442
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 150, ( :1/2), 1)(2,\ 150,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.79123678970.7912367897
L(12)L(\frac12) ≈\approx 0.79123678970.7912367897
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+T 1 + T
3 1+T 1 + T
5 1 1
good7 1−2T+pT2 1 - 2 T + p T^{2}
11 1−2T+pT2 1 - 2 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+pT2 1 + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1−2T+pT2 1 - 2 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1+4T+pT2 1 + 4 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

−12.84190636059108486195039921806, −11.61114212468973479325694460077, −11.12199065028595017208430791111, −10.04002787682700509865642496375, −8.831748546434365307733167695606, −7.907415924764968545249330860332, −6.60420630283526755468874076032, −5.54667504701134155415335928612, −3.85245454580381340504020564721, −1.48708143658565903074344661320, 1.48708143658565903074344661320, 3.85245454580381340504020564721, 5.54667504701134155415335928612, 6.60420630283526755468874076032, 7.907415924764968545249330860332, 8.831748546434365307733167695606, 10.04002787682700509865642496375, 11.12199065028595017208430791111, 11.61114212468973479325694460077, 12.84190636059108486195039921806

Graph of the ZZ-function along the critical line