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Properties

Label 2-29988-1.1-c1-0-14
Degree 22
Conductor 2998829988
Sign 11
Analytic cond. 239.455239.455
Root an. cond. 15.474315.4743
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 11-s − 3·13-s − 17-s + 2·19-s − 4·23-s + 11·25-s − 8·31-s + 8·37-s + 10·43-s − 10·47-s − 3·53-s − 4·55-s − 14·59-s + 8·61-s − 12·65-s − 10·67-s + 5·71-s + 16·73-s + 11·79-s + 12·83-s − 4·85-s + 9·89-s + 8·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.301·11-s − 0.832·13-s − 0.242·17-s + 0.458·19-s − 0.834·23-s + 11/5·25-s − 1.43·31-s + 1.31·37-s + 1.52·43-s − 1.45·47-s − 0.412·53-s − 0.539·55-s − 1.82·59-s + 1.02·61-s − 1.48·65-s − 1.22·67-s + 0.593·71-s + 1.87·73-s + 1.23·79-s + 1.31·83-s − 0.433·85-s + 0.953·89-s + 0.820·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

Λ(s)=(29988s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(29988s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 29988 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2998829988    =    22⋅32⋅72⋅172^{2} \cdot 3^{2} \cdot 7^{2} \cdot 17
Sign: 11
Analytic conductor: 239.455239.455
Root analytic conductor: 15.474315.4743
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 29988, ( :1/2), 1)(2,\ 29988,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.0256275723.025627572
L(12)L(\frac12) ≈\approx 3.0256275723.025627572
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
7 1 1
17 1+T 1 + T
good5 1−4T+pT2 1 - 4 T + p T^{2}
11 1+T+pT2 1 + T + p T^{2}
13 1+3T+pT2 1 + 3 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1+4T+pT2 1 + 4 T + p T^{2}
29 1+pT2 1 + p T^{2}
31 1+8T+pT2 1 + 8 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1−10T+pT2 1 - 10 T + p T^{2}
47 1+10T+pT2 1 + 10 T + p T^{2}
53 1+3T+pT2 1 + 3 T + p T^{2}
59 1+14T+pT2 1 + 14 T + p T^{2}
61 1−8T+pT2 1 - 8 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1−5T+pT2 1 - 5 T + p T^{2}
73 1−16T+pT2 1 - 16 T + p T^{2}
79 1−11T+pT2 1 - 11 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−9T+pT2 1 - 9 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p T^{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

−14.97906750049292, −14.45797478572740, −14.11486558550200, −13.57764667781177, −13.04986299308015, −12.61935037757258, −12.11297272279606, −11.23102842626176, −10.79337440609267, −10.21444372384678, −9.590211435989144, −9.416678510707384, −8.845538785905183, −7.842397063368304, −7.573718520909425, −6.621550420889357, −6.239542370782806, −5.633631858226494, −5.106012749006491, −4.575727799891536, −3.595112932860661, −2.782291447980040, −2.173551089023896, −1.707625068706323, −0.6456610334919439, 0.6456610334919439, 1.707625068706323, 2.173551089023896, 2.782291447980040, 3.595112932860661, 4.575727799891536, 5.106012749006491, 5.633631858226494, 6.239542370782806, 6.621550420889357, 7.573718520909425, 7.842397063368304, 8.845538785905183, 9.416678510707384, 9.590211435989144, 10.21444372384678, 10.79337440609267, 11.23102842626176, 12.11297272279606, 12.61935037757258, 13.04986299308015, 13.57764667781177, 14.11486558550200, 14.45797478572740, 14.97906750049292

Graph of the ZZ-function along the critical line