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Properties

Label 2-4896-1.1-c1-0-52
Degree 22
Conductor 48964896
Sign −1-1
Analytic cond. 39.094739.0947
Root an. cond. 6.252586.25258
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  − 2.96·5-s + 4.15·7-s − 3.19·11-s − 5.35·13-s − 17-s + 1.61·19-s + 8.46·23-s + 3.77·25-s + 6.96·29-s + 6.54·31-s − 12.3·35-s − 6.96·37-s − 8.70·41-s + 4.31·43-s − 5.92·47-s + 10.2·49-s − 4.70·53-s + 9.46·55-s + 7.53·59-s − 10.1·61-s + 15.8·65-s − 3.22·67-s − 13.7·71-s + 5.22·73-s − 13.2·77-s − 13.2·79-s + 4.31·83-s + ⋯
L(s)  = 1  − 1.32·5-s + 1.57·7-s − 0.963·11-s − 1.48·13-s − 0.242·17-s + 0.369·19-s + 1.76·23-s + 0.755·25-s + 1.29·29-s + 1.17·31-s − 2.08·35-s − 1.14·37-s − 1.35·41-s + 0.657·43-s − 0.864·47-s + 1.46·49-s − 0.645·53-s + 1.27·55-s + 0.981·59-s − 1.30·61-s + 1.96·65-s − 0.393·67-s − 1.63·71-s + 0.611·73-s − 1.51·77-s − 1.49·79-s + 0.473·83-s + ⋯

Functional equation

Λ(s)=(4896s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4896s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 48964896    =    25⋅32⋅172^{5} \cdot 3^{2} \cdot 17
Sign: −1-1
Analytic conductor: 39.094739.0947
Root analytic conductor: 6.252586.25258
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4896, ( :1/2), −1)(2,\ 4896,\ (\ :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 1
3 1 1
17 1+T 1 + T
good5 1+2.96T+5T2 1 + 2.96T + 5T^{2}
7 1−4.15T+7T2 1 - 4.15T + 7T^{2}
11 1+3.19T+11T2 1 + 3.19T + 11T^{2}
13 1+5.35T+13T2 1 + 5.35T + 13T^{2}
19 1−1.61T+19T2 1 - 1.61T + 19T^{2}
23 1−8.46T+23T2 1 - 8.46T + 23T^{2}
29 1−6.96T+29T2 1 - 6.96T + 29T^{2}
31 1−6.54T+31T2 1 - 6.54T + 31T^{2}
37 1+6.96T+37T2 1 + 6.96T + 37T^{2}
41 1+8.70T+41T2 1 + 8.70T + 41T^{2}
43 1−4.31T+43T2 1 - 4.31T + 43T^{2}
47 1+5.92T+47T2 1 + 5.92T + 47T^{2}
53 1+4.70T+53T2 1 + 4.70T + 53T^{2}
59 1−7.53T+59T2 1 - 7.53T + 59T^{2}
61 1+10.1T+61T2 1 + 10.1T + 61T^{2}
67 1+3.22T+67T2 1 + 3.22T + 67T^{2}
71 1+13.7T+71T2 1 + 13.7T + 71T^{2}
73 1−5.22T+73T2 1 - 5.22T + 73T^{2}
79 1+13.2T+79T2 1 + 13.2T + 79T^{2}
83 1−4.31T+83T2 1 - 4.31T + 83T^{2}
89 1−7.27T+89T2 1 - 7.27T + 89T^{2}
97 1+2.77T+97T2 1 + 2.77T + 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

−7.941809484926371612945198438187, −7.34367487338673502671339679942, −6.78346169909090632775112850447, −5.33481479788559907253714184498, −4.79191300904464210334878864030, −4.51863174312756331143547324655, −3.21730293777465648369628975693, −2.52943585135309431949769048058, −1.27939297748265963050871482163, 0, 1.27939297748265963050871482163, 2.52943585135309431949769048058, 3.21730293777465648369628975693, 4.51863174312756331143547324655, 4.79191300904464210334878864030, 5.33481479788559907253714184498, 6.78346169909090632775112850447, 7.34367487338673502671339679942, 7.941809484926371612945198438187

Graph of the ZZ-function along the critical line