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Properties

Label 2-8450-1.1-c1-0-240
Degree 22
Conductor 84508450
Sign −1-1
Analytic cond. 67.473567.4735
Root an. cond. 8.214238.21423
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.89·3-s + 4-s + 2.89·6-s − 4.38·7-s + 8-s + 5.38·9-s − 2·11-s + 2.89·12-s − 4.38·14-s + 16-s − 5.86·17-s + 5.38·18-s + 0.973·19-s − 12.6·21-s − 2·22-s − 7.79·23-s + 2.89·24-s + 6.89·27-s − 4.38·28-s + 0.973·29-s + 1.79·31-s + 32-s − 5.79·33-s − 5.86·34-s + 5.38·36-s − 0.591·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.67·3-s + 0.5·4-s + 1.18·6-s − 1.65·7-s + 0.353·8-s + 1.79·9-s − 0.603·11-s + 0.835·12-s − 1.17·14-s + 0.250·16-s − 1.42·17-s + 1.26·18-s + 0.223·19-s − 2.76·21-s − 0.426·22-s − 1.62·23-s + 0.590·24-s + 1.32·27-s − 0.828·28-s + 0.180·29-s + 0.321·31-s + 0.176·32-s − 1.00·33-s − 1.00·34-s + 0.896·36-s − 0.0972·37-s + ⋯

Functional equation

Λ(s)=(8450s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(8450s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 84508450    =    2⋅52⋅1322 \cdot 5^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 67.473567.4735
Root analytic conductor: 8.214238.21423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 8450, ( :1/2), −1)(2,\ 8450,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
5 1 1
13 1 1
good3 1−2.89T+3T2 1 - 2.89T + 3T^{2}
7 1+4.38T+7T2 1 + 4.38T + 7T^{2}
11 1+2T+11T2 1 + 2T + 11T^{2}
17 1+5.86T+17T2 1 + 5.86T + 17T^{2}
19 1−0.973T+19T2 1 - 0.973T + 19T^{2}
23 1+7.79T+23T2 1 + 7.79T + 23T^{2}
29 1−0.973T+29T2 1 - 0.973T + 29T^{2}
31 1−1.79T+31T2 1 - 1.79T + 31T^{2}
37 1+0.591T+37T2 1 + 0.591T + 37T^{2}
41 1+4.81T+41T2 1 + 4.81T + 41T^{2}
43 1+4.68T+43T2 1 + 4.68T + 43T^{2}
47 1−0.381T+47T2 1 - 0.381T + 47T^{2}
53 1+7.79T+53T2 1 + 7.79T + 53T^{2}
59 1−0.973T+59T2 1 - 0.973T + 59T^{2}
61 1+0.817T+61T2 1 + 0.817T + 61T^{2}
67 1−1.79T+67T2 1 - 1.79T + 67T^{2}
71 1−3.92T+71T2 1 - 3.92T + 71T^{2}
73 1+6T+73T2 1 + 6T + 73T^{2}
79 1−10.9T+79T2 1 - 10.9T + 79T^{2}
83 1+6.97T+83T2 1 + 6.97T + 83T^{2}
89 1+0.973T+89T2 1 + 0.973T + 89T^{2}
97 1+18.6T+97T2 1 + 18.6T + 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

−7.48056836847830329810809883224, −6.63400954704746555889226548186, −6.34622676821150826633455054453, −5.27552121438622773729162193973, −4.27492596803446109252984822230, −3.76529279397457404677385463441, −3.04879631076630862538005337193, −2.55211502194329041332436690479, −1.80262589868690942155437903756, 0, 1.80262589868690942155437903756, 2.55211502194329041332436690479, 3.04879631076630862538005337193, 3.76529279397457404677385463441, 4.27492596803446109252984822230, 5.27552121438622773729162193973, 6.34622676821150826633455054453, 6.63400954704746555889226548186, 7.48056836847830329810809883224

Graph of the ZZ-function along the critical line