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Properties

Label 2-315-1.1-c1-0-3
Degree 22
Conductor 315315
Sign 11
Analytic cond. 2.515282.51528
Root an. cond. 1.585961.58596
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  − 2·4-s + 5-s + 7-s + 3·11-s + 5·13-s + 4·16-s − 3·17-s + 2·19-s − 2·20-s + 6·23-s + 25-s − 2·28-s − 3·29-s − 4·31-s + 35-s + 2·37-s + 12·41-s − 10·43-s − 6·44-s − 9·47-s + 49-s − 10·52-s − 12·53-s + 3·55-s + 8·61-s − 8·64-s + 5·65-s + ⋯
L(s)  = 1  − 4-s + 0.447·5-s + 0.377·7-s + 0.904·11-s + 1.38·13-s + 16-s − 0.727·17-s + 0.458·19-s − 0.447·20-s + 1.25·23-s + 1/5·25-s − 0.377·28-s − 0.557·29-s − 0.718·31-s + 0.169·35-s + 0.328·37-s + 1.87·41-s − 1.52·43-s − 0.904·44-s − 1.31·47-s + 1/7·49-s − 1.38·52-s − 1.64·53-s + 0.404·55-s + 1.02·61-s − 64-s + 0.620·65-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 2.515282.51528
Root analytic conductor: 1.585961.58596
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 315, ( :1/2), 1)(2,\ 315,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2730829061.273082906
L(12)L(\frac12) ≈\approx 1.2730829061.273082906
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)
bad3 1 1
5 1−T 1 - T
7 1−T 1 - T
good2 1+pT2 1 + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+3T+pT2 1 + 3 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−12T+pT2 1 - 12 T + p T^{2}
43 1+10T+pT2 1 + 10 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1+12T+pT2 1 + 12 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1+12T+pT2 1 + 12 T + p T^{2}
89 1−12T+pT2 1 - 12 T + p T^{2}
97 1+T+pT2 1 + 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

−11.50769791186352205274303435148, −10.81642134200306905635056067311, −9.502060864672765915422726581101, −9.001872818363131182920485953111, −8.087023964728641191414598204496, −6.69771158952637091109528432278, −5.63914839037632160871019437372, −4.54321060660836499326534703030, −3.45067056060882590557565434961, −1.35738749247200780270220785705, 1.35738749247200780270220785705, 3.45067056060882590557565434961, 4.54321060660836499326534703030, 5.63914839037632160871019437372, 6.69771158952637091109528432278, 8.087023964728641191414598204496, 9.001872818363131182920485953111, 9.502060864672765915422726581101, 10.81642134200306905635056067311, 11.50769791186352205274303435148

Graph of the ZZ-function along the critical line