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Properties

Label 2-5408-1.1-c1-0-99
Degree 22
Conductor 54085408
Sign 11
Analytic cond. 43.183043.1830
Root an. cond. 6.571386.57138
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  + 2.96·3-s + 1.23·5-s + 4.80·7-s + 5.80·9-s − 4.80·11-s + 3.66·15-s − 17-s + 5.13·19-s + 14.2·21-s − 3.96·23-s − 3.47·25-s + 8.32·27-s + 1.33·29-s + 3.66·31-s − 14.2·33-s + 5.93·35-s − 3.62·37-s + 7.66·41-s − 2.38·43-s + 7.16·45-s + 5.94·47-s + 16.0·49-s − 2.96·51-s + 0.139·53-s − 5.93·55-s + 15.2·57-s + 5.47·59-s + ⋯
L(s)  = 1  + 1.71·3-s + 0.552·5-s + 1.81·7-s + 1.93·9-s − 1.44·11-s + 0.946·15-s − 0.242·17-s + 1.17·19-s + 3.11·21-s − 0.825·23-s − 0.694·25-s + 1.60·27-s + 0.247·29-s + 0.658·31-s − 2.48·33-s + 1.00·35-s − 0.595·37-s + 1.19·41-s − 0.364·43-s + 1.06·45-s + 0.867·47-s + 2.29·49-s − 0.415·51-s + 0.0191·53-s − 0.800·55-s + 2.01·57-s + 0.712·59-s + ⋯

Functional equation

Λ(s)=(5408s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5408s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 54085408    =    25⋅1322^{5} \cdot 13^{2}
Sign: 11
Analytic conductor: 43.183043.1830
Root analytic conductor: 6.571386.57138
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5408, ( :1/2), 1)(2,\ 5408,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.1375542725.137554272
L(12)L(\frac12) ≈\approx 5.1375542725.137554272
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 1
13 1 1
good3 1−2.96T+3T2 1 - 2.96T + 3T^{2}
5 1−1.23T+5T2 1 - 1.23T + 5T^{2}
7 1−4.80T+7T2 1 - 4.80T + 7T^{2}
11 1+4.80T+11T2 1 + 4.80T + 11T^{2}
17 1+T+17T2 1 + T + 17T^{2}
19 1−5.13T+19T2 1 - 5.13T + 19T^{2}
23 1+3.96T+23T2 1 + 3.96T + 23T^{2}
29 1−1.33T+29T2 1 - 1.33T + 29T^{2}
31 1−3.66T+31T2 1 - 3.66T + 31T^{2}
37 1+3.62T+37T2 1 + 3.62T + 37T^{2}
41 1−7.66T+41T2 1 - 7.66T + 41T^{2}
43 1+2.38T+43T2 1 + 2.38T + 43T^{2}
47 1−5.94T+47T2 1 - 5.94T + 47T^{2}
53 1−0.139T+53T2 1 - 0.139T + 53T^{2}
59 1−5.47T+59T2 1 - 5.47T + 59T^{2}
61 1−4.94T+61T2 1 - 4.94T + 61T^{2}
67 1+6.74T+67T2 1 + 6.74T + 67T^{2}
71 1+1.13T+71T2 1 + 1.13T + 71T^{2}
73 1−5.27T+73T2 1 - 5.27T + 73T^{2}
79 1+4.77T+79T2 1 + 4.77T + 79T^{2}
83 1+1.60T+83T2 1 + 1.60T + 83T^{2}
89 1−13.2T+89T2 1 - 13.2T + 89T^{2}
97 1−15.2T+97T2 1 - 15.2T + 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

−8.102098829400259032741422226381, −7.73848525305371960471728038311, −7.21601879929890544657723196023, −5.85574625251574074579957445111, −5.13746139800441031827460045007, −4.46892552812076689336983611688, −3.59264457097128764747637843461, −2.50739878676306923487312739145, −2.19134061192460131251088737841, −1.24631448189815198334064489324, 1.24631448189815198334064489324, 2.19134061192460131251088737841, 2.50739878676306923487312739145, 3.59264457097128764747637843461, 4.46892552812076689336983611688, 5.13746139800441031827460045007, 5.85574625251574074579957445111, 7.21601879929890544657723196023, 7.73848525305371960471728038311, 8.102098829400259032741422226381

Graph of the ZZ-function along the critical line