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Properties

Label 2-297-1.1-c1-0-6
Degree 22
Conductor 297297
Sign −1-1
Analytic cond. 2.371552.37155
Root an. cond. 1.539981.53998
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  − 2.73·2-s + 5.46·4-s + 0.732·5-s − 3.73·7-s − 9.46·8-s − 2·10-s − 11-s − 0.267·13-s + 10.1·14-s + 14.9·16-s + 4.19·17-s − 5.19·19-s + 4·20-s + 2.73·22-s − 8·23-s − 4.46·25-s + 0.732·26-s − 20.3·28-s + 1.26·29-s + 0.535·31-s − 21.8·32-s − 11.4·34-s − 2.73·35-s − 6.46·37-s + 14.1·38-s − 6.92·40-s − 1.46·41-s + ⋯
L(s)  = 1  − 1.93·2-s + 2.73·4-s + 0.327·5-s − 1.41·7-s − 3.34·8-s − 0.632·10-s − 0.301·11-s − 0.0743·13-s + 2.72·14-s + 3.73·16-s + 1.01·17-s − 1.19·19-s + 0.894·20-s + 0.582·22-s − 1.66·23-s − 0.892·25-s + 0.143·26-s − 3.85·28-s + 0.235·29-s + 0.0962·31-s − 3.86·32-s − 1.96·34-s − 0.461·35-s − 1.06·37-s + 2.30·38-s − 1.09·40-s − 0.228·41-s + ⋯

Functional equation

Λ(s)=(297s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(297s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 297297    =    33⋅113^{3} \cdot 11
Sign: −1-1
Analytic conductor: 2.371552.37155
Root analytic conductor: 1.539981.53998
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 297, ( :1/2), −1)(2,\ 297,\ (\ :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)
bad3 1 1
11 1+T 1 + T
good2 1+2.73T+2T2 1 + 2.73T + 2T^{2}
5 1−0.732T+5T2 1 - 0.732T + 5T^{2}
7 1+3.73T+7T2 1 + 3.73T + 7T^{2}
13 1+0.267T+13T2 1 + 0.267T + 13T^{2}
17 1−4.19T+17T2 1 - 4.19T + 17T^{2}
19 1+5.19T+19T2 1 + 5.19T + 19T^{2}
23 1+8T+23T2 1 + 8T + 23T^{2}
29 1−1.26T+29T2 1 - 1.26T + 29T^{2}
31 1−0.535T+31T2 1 - 0.535T + 31T^{2}
37 1+6.46T+37T2 1 + 6.46T + 37T^{2}
41 1+1.46T+41T2 1 + 1.46T + 41T^{2}
43 1−3.46T+43T2 1 - 3.46T + 43T^{2}
47 1+10.1T+47T2 1 + 10.1T + 47T^{2}
53 1+4.73T+53T2 1 + 4.73T + 53T^{2}
59 1+10.1T+59T2 1 + 10.1T + 59T^{2}
61 1+4.26T+61T2 1 + 4.26T + 61T^{2}
67 1−5.92T+67T2 1 - 5.92T + 67T^{2}
71 1−13.8T+71T2 1 - 13.8T + 71T^{2}
73 1−3.19T+73T2 1 - 3.19T + 73T^{2}
79 1+5.73T+79T2 1 + 5.73T + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1−0.339T+89T2 1 - 0.339T + 89T^{2}
97 1+5.39T+97T2 1 + 5.39T + 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

−10.82064368787251089401295184476, −9.922946070683342917394260083375, −9.699982788297855978122388060119, −8.511700238161327440662268732347, −7.69953954818979481840226331198, −6.55397274843630290556864924370, −5.94520275907457442358055304251, −3.35149336968103963333524756878, −2.02172474427898299129359430859, 0, 2.02172474427898299129359430859, 3.35149336968103963333524756878, 5.94520275907457442358055304251, 6.55397274843630290556864924370, 7.69953954818979481840226331198, 8.511700238161327440662268732347, 9.699982788297855978122388060119, 9.922946070683342917394260083375, 10.82064368787251089401295184476

Graph of the ZZ-function along the critical line