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Properties

Label 2-525-1.1-c1-0-3
Degree 22
Conductor 525525
Sign 11
Analytic cond. 4.192144.19214
Root an. cond. 2.047472.04747
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.30·2-s + 3-s − 0.302·4-s − 1.30·6-s + 7-s + 3·8-s + 9-s − 3·11-s − 0.302·12-s + 4.60·13-s − 1.30·14-s − 3.30·16-s − 2.60·17-s − 1.30·18-s − 0.605·19-s + 21-s + 3.90·22-s + 8.21·23-s + 3·24-s − 6·26-s + 27-s − 0.302·28-s − 0.394·29-s + 7.21·31-s − 1.69·32-s − 3·33-s + 3.39·34-s + ⋯
L(s)  = 1  − 0.921·2-s + 0.577·3-s − 0.151·4-s − 0.531·6-s + 0.377·7-s + 1.06·8-s + 0.333·9-s − 0.904·11-s − 0.0874·12-s + 1.27·13-s − 0.348·14-s − 0.825·16-s − 0.631·17-s − 0.307·18-s − 0.138·19-s + 0.218·21-s + 0.833·22-s + 1.71·23-s + 0.612·24-s − 1.17·26-s + 0.192·27-s − 0.0572·28-s − 0.0732·29-s + 1.29·31-s − 0.300·32-s − 0.522·33-s + 0.582·34-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 4.192144.19214
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 525, ( :1/2), 1)(2,\ 525,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.0509803851.050980385
L(12)L(\frac12) ≈\approx 1.0509803851.050980385
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−T 1 - T
5 1 1
7 1−T 1 - T
good2 1+1.30T+2T2 1 + 1.30T + 2T^{2}
11 1+3T+11T2 1 + 3T + 11T^{2}
13 1−4.60T+13T2 1 - 4.60T + 13T^{2}
17 1+2.60T+17T2 1 + 2.60T + 17T^{2}
19 1+0.605T+19T2 1 + 0.605T + 19T^{2}
23 1−8.21T+23T2 1 - 8.21T + 23T^{2}
29 1+0.394T+29T2 1 + 0.394T + 29T^{2}
31 1−7.21T+31T2 1 - 7.21T + 31T^{2}
37 1−10.2T+37T2 1 - 10.2T + 37T^{2}
41 1+41T2 1 + 41T^{2}
43 1−2.39T+43T2 1 - 2.39T + 43T^{2}
47 1−3.39T+47T2 1 - 3.39T + 47T^{2}
53 1+11.2T+53T2 1 + 11.2T + 53T^{2}
59 1+3.39T+59T2 1 + 3.39T + 59T^{2}
61 1−13.2T+61T2 1 - 13.2T + 61T^{2}
67 1−8.39T+67T2 1 - 8.39T + 67T^{2}
71 1+3T+71T2 1 + 3T + 71T^{2}
73 1+6.60T+73T2 1 + 6.60T + 73T^{2}
79 1−6.81T+79T2 1 - 6.81T + 79T^{2}
83 1−11.2T+83T2 1 - 11.2T + 83T^{2}
89 1+13.8T+89T2 1 + 13.8T + 89T^{2}
97 1+15.2T+97T2 1 + 15.2T + 97T^{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

−10.79780919306747793069880154372, −9.793335438532053116002002357886, −8.966703555107316466281600066345, −8.318211282096011046886671201429, −7.67393818503279158702144974855, −6.54179657572404628384987199966, −5.10049332802200492366671889537, −4.12608966068775967928571524911, −2.65940509197039901278093412004, −1.12404555162790709900654244864, 1.12404555162790709900654244864, 2.65940509197039901278093412004, 4.12608966068775967928571524911, 5.10049332802200492366671889537, 6.54179657572404628384987199966, 7.67393818503279158702144974855, 8.318211282096011046886671201429, 8.966703555107316466281600066345, 9.793335438532053116002002357886, 10.79780919306747793069880154372

Graph of the ZZ-function along the critical line