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Properties

Label 2-1638-1.1-c1-0-17
Degree 22
Conductor 16381638
Sign −1-1
Analytic cond. 13.079413.0794
Root an. cond. 3.616553.61655
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 + 4-s − 3.37·5-s − 7-s − 8-s + 3.37·10-s + 3.37·11-s − 13-s + 14-s + 16-s + 1.37·17-s + 7.37·19-s − 3.37·20-s − 3.37·22-s − 9.37·23-s + 6.37·25-s + 26-s − 28-s − 0.627·29-s + 6.74·31-s − 32-s − 1.37·34-s + 3.37·35-s + 1.37·37-s − 7.37·38-s + 3.37·40-s + 2.74·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.50·5-s − 0.377·7-s − 0.353·8-s + 1.06·10-s + 1.01·11-s − 0.277·13-s + 0.267·14-s + 0.250·16-s + 0.332·17-s + 1.69·19-s − 0.754·20-s − 0.718·22-s − 1.95·23-s + 1.27·25-s + 0.196·26-s − 0.188·28-s − 0.116·29-s + 1.21·31-s − 0.176·32-s − 0.235·34-s + 0.570·35-s + 0.225·37-s − 1.19·38-s + 0.533·40-s + 0.428·41-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: −1-1
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1638, ( :1/2), −1)(2,\ 1638,\ (\ :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
3 1 1
7 1+T 1 + T
13 1+T 1 + T
good5 1+3.37T+5T2 1 + 3.37T + 5T^{2}
11 1−3.37T+11T2 1 - 3.37T + 11T^{2}
17 1−1.37T+17T2 1 - 1.37T + 17T^{2}
19 1−7.37T+19T2 1 - 7.37T + 19T^{2}
23 1+9.37T+23T2 1 + 9.37T + 23T^{2}
29 1+0.627T+29T2 1 + 0.627T + 29T^{2}
31 1−6.74T+31T2 1 - 6.74T + 31T^{2}
37 1−1.37T+37T2 1 - 1.37T + 37T^{2}
41 1−2.74T+41T2 1 - 2.74T + 41T^{2}
43 1+6.11T+43T2 1 + 6.11T + 43T^{2}
47 1+12.7T+47T2 1 + 12.7T + 47T^{2}
53 1−2.74T+53T2 1 - 2.74T + 53T^{2}
59 1+2T+59T2 1 + 2T + 59T^{2}
61 1+12.1T+61T2 1 + 12.1T + 61T^{2}
67 1+13.4T+67T2 1 + 13.4T + 67T^{2}
71 1+6.74T+71T2 1 + 6.74T + 71T^{2}
73 1+2.62T+73T2 1 + 2.62T + 73T^{2}
79 1+6.74T+79T2 1 + 6.74T + 79T^{2}
83 1−6T+83T2 1 - 6T + 83T^{2}
89 1−14.7T+89T2 1 - 14.7T + 89T^{2}
97 1+15.4T+97T2 1 + 15.4T + 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

−8.987704821613918900132439242807, −7.971201297115256601396982665464, −7.70656538507678598770762637755, −6.77507242625469800641900267219, −5.96831913046664675574752327681, −4.64992976034127445628747307556, −3.74282880291515877863947741562, −3.00834552043087154785451106595, −1.35575343499617005022179923724, 0, 1.35575343499617005022179923724, 3.00834552043087154785451106595, 3.74282880291515877863947741562, 4.64992976034127445628747307556, 5.96831913046664675574752327681, 6.77507242625469800641900267219, 7.70656538507678598770762637755, 7.971201297115256601396982665464, 8.987704821613918900132439242807

Graph of the ZZ-function along the critical line