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Properties

Label 2-9075-1.1-c1-0-12
Degree 22
Conductor 90759075
Sign 11
Analytic cond. 72.464272.4642
Root an. cond. 8.512598.51259
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  − 0.792·2-s − 3-s − 1.37·4-s + 0.792·6-s − 2.52·7-s + 2.67·8-s + 9-s + 1.37·12-s − 6.78·13-s + 2·14-s + 0.627·16-s − 6.63·17-s − 0.792·18-s + 7.72·19-s + 2.52·21-s + 8·23-s − 2.67·24-s + 5.37·26-s − 27-s + 3.46·28-s + 3.16·29-s − 3.37·31-s − 5.84·32-s + 5.25·34-s − 1.37·36-s + 5·37-s − 6.11·38-s + ⋯
L(s)  = 1  − 0.560·2-s − 0.577·3-s − 0.686·4-s + 0.323·6-s − 0.954·7-s + 0.944·8-s + 0.333·9-s + 0.396·12-s − 1.88·13-s + 0.534·14-s + 0.156·16-s − 1.60·17-s − 0.186·18-s + 1.77·19-s + 0.550·21-s + 1.66·23-s − 0.545·24-s + 1.05·26-s − 0.192·27-s + 0.654·28-s + 0.588·29-s − 0.605·31-s − 1.03·32-s + 0.901·34-s − 0.228·36-s + 0.821·37-s − 0.992·38-s + ⋯

Functional equation

Λ(s)=(9075s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9075s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 90759075    =    3⋅52⋅1123 \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 72.464272.4642
Root analytic conductor: 8.512598.51259
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9075, ( :1/2), 1)(2,\ 9075,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.30240223560.3024022356
L(12)L(\frac12) ≈\approx 0.30240223560.3024022356
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
11 1 1
good2 1+0.792T+2T2 1 + 0.792T + 2T^{2}
7 1+2.52T+7T2 1 + 2.52T + 7T^{2}
13 1+6.78T+13T2 1 + 6.78T + 13T^{2}
17 1+6.63T+17T2 1 + 6.63T + 17T^{2}
19 1−7.72T+19T2 1 - 7.72T + 19T^{2}
23 1−8T+23T2 1 - 8T + 23T^{2}
29 1−3.16T+29T2 1 - 3.16T + 29T^{2}
31 1+3.37T+31T2 1 + 3.37T + 31T^{2}
37 1−5T+37T2 1 - 5T + 37T^{2}
41 1+6.92T+41T2 1 + 6.92T + 41T^{2}
43 1+9.94T+43T2 1 + 9.94T + 43T^{2}
47 1−2.74T+47T2 1 - 2.74T + 47T^{2}
53 1+12.7T+53T2 1 + 12.7T + 53T^{2}
59 1−4T+59T2 1 - 4T + 59T^{2}
61 1+2.67T+61T2 1 + 2.67T + 61T^{2}
67 1−6.11T+67T2 1 - 6.11T + 67T^{2}
71 1+0.744T+71T2 1 + 0.744T + 71T^{2}
73 1+5.19T+73T2 1 + 5.19T + 73T^{2}
79 1−0.147T+79T2 1 - 0.147T + 79T^{2}
83 1−1.87T+83T2 1 - 1.87T + 83T^{2}
89 1+17.4T+89T2 1 + 17.4T + 89T^{2}
97 1+3.62T+97T2 1 + 3.62T + 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

−7.64463086674054385348181414628, −6.97313359155605649135322374470, −6.69645395729456967319265324049, −5.48369505572690581021023959685, −4.94886801440622726841952345873, −4.50333954128640190777380997918, −3.37930402626546639070943278359, −2.66578267326586907416951650500, −1.43197190776712929730520281370, −0.31297446947974179927242880302, 0.31297446947974179927242880302, 1.43197190776712929730520281370, 2.66578267326586907416951650500, 3.37930402626546639070943278359, 4.50333954128640190777380997918, 4.94886801440622726841952345873, 5.48369505572690581021023959685, 6.69645395729456967319265324049, 6.97313359155605649135322374470, 7.64463086674054385348181414628

Graph of the ZZ-function along the critical line