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Properties

Label 2-510-1.1-c1-0-4
Degree 22
Conductor 510510
Sign 11
Analytic cond. 4.072374.07237
Root an. cond. 2.018012.01801
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 − 5-s − 6-s + 2·7-s + 8-s + 9-s − 10-s − 12-s + 4·13-s + 2·14-s + 15-s + 16-s − 17-s + 18-s + 4·19-s − 20-s − 2·21-s + 4·23-s − 24-s + 25-s + 4·26-s − 27-s + 2·28-s + 2·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 1.10·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.436·21-s + 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s + 0.371·29-s + 0.182·30-s + ⋯

Functional equation

Λ(s)=(510s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(510s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 510 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 510510    =    2⋅3⋅5⋅172 \cdot 3 \cdot 5 \cdot 17
Sign: 11
Analytic conductor: 4.072374.07237
Root analytic conductor: 2.018012.01801
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 510, ( :1/2), 1)(2,\ 510,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9361123181.936112318
L(12)L(\frac12) ≈\approx 1.9361123181.936112318
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+T 1 + T
17 1+T 1 + T
good7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1−2T+pT2 1 - 2 T + p T^{2}
31 1+pT2 1 + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+4T+pT2 1 + 4 T + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1−2T+pT2 1 - 2 T + p T^{2}
59 1+2T+pT2 1 + 2 T + p T^{2}
61 1+14T+pT2 1 + 14 T + p T^{2}
67 1−2T+pT2 1 - 2 T + p T^{2}
71 1+6T+pT2 1 + 6 T + p T^{2}
73 1+4T+pT2 1 + 4 T + p T^{2}
79 1+12T+pT2 1 + 12 T + p T^{2}
83 1−8T+pT2 1 - 8 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

−11.17937595466176513744449870847, −10.40568418942802751264406629690, −9.065035379514449228234893396125, −8.037143896449908332265909173396, −7.13577035934415708186454315215, −6.13674211142237372481129150645, −5.16788486225288439800302517751, −4.32574622516919833254834247273, −3.16587562444602674822441472468, −1.37173249517559434715622111936, 1.37173249517559434715622111936, 3.16587562444602674822441472468, 4.32574622516919833254834247273, 5.16788486225288439800302517751, 6.13674211142237372481129150645, 7.13577035934415708186454315215, 8.037143896449908332265909173396, 9.065035379514449228234893396125, 10.40568418942802751264406629690, 11.17937595466176513744449870847

Graph of the ZZ-function along the critical line