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Properties

Label 2-99e2-1.1-c1-0-305
Degree 22
Conductor 98019801
Sign 11
Analytic cond. 78.261378.2613
Root an. cond. 8.846548.84654
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  + 2.71·2-s + 5.39·4-s + 3.51·5-s − 3.83·7-s + 9.22·8-s + 9.54·10-s − 1.15·13-s − 10.4·14-s + 14.3·16-s + 5.40·17-s + 0.558·19-s + 18.9·20-s − 2.75·23-s + 7.32·25-s − 3.12·26-s − 20.6·28-s − 4.70·29-s + 8.76·31-s + 20.4·32-s + 14.6·34-s − 13.4·35-s − 4.31·37-s + 1.51·38-s + 32.3·40-s + 0.476·41-s + 3.24·43-s − 7.48·46-s + ⋯
L(s)  = 1  + 1.92·2-s + 2.69·4-s + 1.56·5-s − 1.44·7-s + 3.26·8-s + 3.01·10-s − 0.319·13-s − 2.78·14-s + 3.57·16-s + 1.31·17-s + 0.128·19-s + 4.23·20-s − 0.574·23-s + 1.46·25-s − 0.613·26-s − 3.90·28-s − 0.874·29-s + 1.57·31-s + 3.61·32-s + 2.52·34-s − 2.27·35-s − 0.709·37-s + 0.246·38-s + 5.12·40-s + 0.0743·41-s + 0.494·43-s − 1.10·46-s + ⋯

Functional equation

Λ(s)=(9801s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9801s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 98019801    =    34⋅1123^{4} \cdot 11^{2}
Sign: 11
Analytic conductor: 78.261378.2613
Root analytic conductor: 8.846548.84654
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9801, ( :1/2), 1)(2,\ 9801,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 9.9518352699.951835269
L(12)L(\frac12) ≈\approx 9.9518352699.951835269
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 1
11 1 1
good2 1−2.71T+2T2 1 - 2.71T + 2T^{2}
5 1−3.51T+5T2 1 - 3.51T + 5T^{2}
7 1+3.83T+7T2 1 + 3.83T + 7T^{2}
13 1+1.15T+13T2 1 + 1.15T + 13T^{2}
17 1−5.40T+17T2 1 - 5.40T + 17T^{2}
19 1−0.558T+19T2 1 - 0.558T + 19T^{2}
23 1+2.75T+23T2 1 + 2.75T + 23T^{2}
29 1+4.70T+29T2 1 + 4.70T + 29T^{2}
31 1−8.76T+31T2 1 - 8.76T + 31T^{2}
37 1+4.31T+37T2 1 + 4.31T + 37T^{2}
41 1−0.476T+41T2 1 - 0.476T + 41T^{2}
43 1−3.24T+43T2 1 - 3.24T + 43T^{2}
47 1−4.42T+47T2 1 - 4.42T + 47T^{2}
53 1−11.2T+53T2 1 - 11.2T + 53T^{2}
59 1−6.79T+59T2 1 - 6.79T + 59T^{2}
61 1+5.53T+61T2 1 + 5.53T + 61T^{2}
67 1−6.70T+67T2 1 - 6.70T + 67T^{2}
71 1−5.88T+71T2 1 - 5.88T + 71T^{2}
73 1+7.06T+73T2 1 + 7.06T + 73T^{2}
79 1−5.84T+79T2 1 - 5.84T + 79T^{2}
83 1−4.80T+83T2 1 - 4.80T + 83T^{2}
89 1+11.1T+89T2 1 + 11.1T + 89T^{2}
97 1+18.1T+97T2 1 + 18.1T + 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

−7.12527306787632070853049163389, −6.69762775099621185048652192472, −5.99787886799053135570753414126, −5.66559694426020150486267327587, −5.15221195108629837199008237344, −4.12464336872101141780216469509, −3.45796282069587645194193429798, −2.73088817251555242432180744253, −2.25163335850922367096809829080, −1.18792169670792347113174024706, 1.18792169670792347113174024706, 2.25163335850922367096809829080, 2.73088817251555242432180744253, 3.45796282069587645194193429798, 4.12464336872101141780216469509, 5.15221195108629837199008237344, 5.66559694426020150486267327587, 5.99787886799053135570753414126, 6.69762775099621185048652192472, 7.12527306787632070853049163389

Graph of the ZZ-function along the critical line