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Properties

Label 2-15e2-1.1-c1-0-4
Degree 22
Conductor 225225
Sign 11
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
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  + 2·2-s + 2·4-s + 3·7-s − 2·11-s − 13-s + 6·14-s − 4·16-s + 2·17-s − 5·19-s − 4·22-s + 6·23-s − 2·26-s + 6·28-s − 10·29-s − 3·31-s − 8·32-s + 4·34-s − 2·37-s − 10·38-s + 8·41-s − 43-s − 4·44-s + 12·46-s + 2·47-s + 2·49-s − 2·52-s − 4·53-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.13·7-s − 0.603·11-s − 0.277·13-s + 1.60·14-s − 16-s + 0.485·17-s − 1.14·19-s − 0.852·22-s + 1.25·23-s − 0.392·26-s + 1.13·28-s − 1.85·29-s − 0.538·31-s − 1.41·32-s + 0.685·34-s − 0.328·37-s − 1.62·38-s + 1.24·41-s − 0.152·43-s − 0.603·44-s + 1.76·46-s + 0.291·47-s + 2/7·49-s − 0.277·52-s − 0.549·53-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 225, ( :1/2), 1)(2,\ 225,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.4113363422.411336342
L(12)L(\frac12) ≈\approx 2.4113363422.411336342
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 1
good2 1−pT+pT2 1 - p T + p T^{2}
7 1−3T+pT2 1 - 3 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
13 1+T+pT2 1 + T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+10T+pT2 1 + 10 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1−8T+pT2 1 - 8 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1−10T+pT2 1 - 10 T + p T^{2}
61 1−7T+pT2 1 - 7 T + p T^{2}
67 1−3T+pT2 1 - 3 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1+pT2 1 + p T^{2}
83 1−6T+pT2 1 - 6 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+17T+pT2 1 + 17 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

−12.58033395895800211021523424017, −11.37636969058177948973372420565, −10.84373752796776504705892968641, −9.294135845295806835405725735271, −8.102126663118456650752904190012, −6.99182605075727565742724929802, −5.60694328736998842121660037613, −4.92213034669592094010306499374, −3.78545707329788062312381068333, −2.27079729708413704097834802658, 2.27079729708413704097834802658, 3.78545707329788062312381068333, 4.92213034669592094010306499374, 5.60694328736998842121660037613, 6.99182605075727565742724929802, 8.102126663118456650752904190012, 9.294135845295806835405725735271, 10.84373752796776504705892968641, 11.37636969058177948973372420565, 12.58033395895800211021523424017

Graph of the ZZ-function along the critical line