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Properties

Label 2-4477-1.1-c1-0-138
Degree 22
Conductor 44774477
Sign −1-1
Analytic cond. 35.749035.7490
Root an. cond. 5.979045.97904
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  − 1.43·2-s − 3.36·3-s + 0.0571·4-s + 1.55·5-s + 4.82·6-s − 3.66·7-s + 2.78·8-s + 8.30·9-s − 2.23·10-s − 0.192·12-s + 3.10·13-s + 5.26·14-s − 5.23·15-s − 4.11·16-s + 4.34·17-s − 11.9·18-s − 2.65·19-s + 0.0889·20-s + 12.3·21-s + 2.08·23-s − 9.36·24-s − 2.57·25-s − 4.45·26-s − 17.8·27-s − 0.209·28-s − 4.07·29-s + 7.50·30-s + ⋯
L(s)  = 1  − 1.01·2-s − 1.94·3-s + 0.0285·4-s + 0.696·5-s + 1.96·6-s − 1.38·7-s + 0.985·8-s + 2.76·9-s − 0.706·10-s − 0.0554·12-s + 0.862·13-s + 1.40·14-s − 1.35·15-s − 1.02·16-s + 1.05·17-s − 2.80·18-s − 0.609·19-s + 0.0198·20-s + 2.69·21-s + 0.435·23-s − 1.91·24-s − 0.515·25-s − 0.874·26-s − 3.42·27-s − 0.0396·28-s − 0.757·29-s + 1.37·30-s + ⋯

Functional equation

Λ(s)=(4477s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4477 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4477s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4477 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 44774477    =    112⋅3711^{2} \cdot 37
Sign: −1-1
Analytic conductor: 35.749035.7490
Root analytic conductor: 5.979045.97904
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4477, ( :1/2), −1)(2,\ 4477,\ (\ :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)
bad11 1 1
37 1−T 1 - T
good2 1+1.43T+2T2 1 + 1.43T + 2T^{2}
3 1+3.36T+3T2 1 + 3.36T + 3T^{2}
5 1−1.55T+5T2 1 - 1.55T + 5T^{2}
7 1+3.66T+7T2 1 + 3.66T + 7T^{2}
13 1−3.10T+13T2 1 - 3.10T + 13T^{2}
17 1−4.34T+17T2 1 - 4.34T + 17T^{2}
19 1+2.65T+19T2 1 + 2.65T + 19T^{2}
23 1−2.08T+23T2 1 - 2.08T + 23T^{2}
29 1+4.07T+29T2 1 + 4.07T + 29T^{2}
31 1+3.76T+31T2 1 + 3.76T + 31T^{2}
41 1−6.47T+41T2 1 - 6.47T + 41T^{2}
43 1−3.77T+43T2 1 - 3.77T + 43T^{2}
47 1+8.91T+47T2 1 + 8.91T + 47T^{2}
53 1+2.24T+53T2 1 + 2.24T + 53T^{2}
59 1−11.8T+59T2 1 - 11.8T + 59T^{2}
61 1+13.9T+61T2 1 + 13.9T + 61T^{2}
67 1+15.3T+67T2 1 + 15.3T + 67T^{2}
71 1−8.76T+71T2 1 - 8.76T + 71T^{2}
73 1−7.43T+73T2 1 - 7.43T + 73T^{2}
79 1+13.3T+79T2 1 + 13.3T + 79T^{2}
83 1−9.93T+83T2 1 - 9.93T + 83T^{2}
89 1−17.7T+89T2 1 - 17.7T + 89T^{2}
97 1+8.23T+97T2 1 + 8.23T + 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

−7.83840739160965623336180246972, −7.16747812310974745583844927519, −6.38278628765974092906584916359, −5.95320453737608094619930471728, −5.32917920830693388854319858008, −4.35338466229133012835257547339, −3.51569998108764043105324516735, −1.82954525331226377733107257264, −0.934118076663053757161614087735, 0, 0.934118076663053757161614087735, 1.82954525331226377733107257264, 3.51569998108764043105324516735, 4.35338466229133012835257547339, 5.32917920830693388854319858008, 5.95320453737608094619930471728, 6.38278628765974092906584916359, 7.16747812310974745583844927519, 7.83840739160965623336180246972

Graph of the ZZ-function along the critical line