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Properties

Label 2-4140-1.1-c1-0-25
Degree 22
Conductor 41404140
Sign −1-1
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 2·7-s + 4·11-s − 2·13-s − 6·17-s + 2·19-s + 23-s + 25-s + 4·29-s + 8·31-s + 2·35-s + 2·37-s − 6·43-s + 4·47-s − 3·49-s − 10·53-s − 4·55-s − 4·59-s − 6·61-s + 2·65-s − 10·67-s + 8·71-s − 10·73-s − 8·77-s − 14·79-s + 4·83-s + 6·85-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 1.20·11-s − 0.554·13-s − 1.45·17-s + 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.742·29-s + 1.43·31-s + 0.338·35-s + 0.328·37-s − 0.914·43-s + 0.583·47-s − 3/7·49-s − 1.37·53-s − 0.539·55-s − 0.520·59-s − 0.768·61-s + 0.248·65-s − 1.22·67-s + 0.949·71-s − 1.17·73-s − 0.911·77-s − 1.57·79-s + 0.439·83-s + 0.650·85-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: −1-1
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4140, ( :1/2), −1)(2,\ 4140,\ (\ :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
5 1+T 1 + T
23 1−T 1 - T
good7 1+2T+pT2 1 + 2 T + p T^{2}
11 1−4T+pT2 1 - 4 T + p T^{2}
13 1+2T+pT2 1 + 2 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1+pT2 1 + p T^{2}
43 1+6T+pT2 1 + 6 T + p T^{2}
47 1−4T+pT2 1 - 4 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1+4T+pT2 1 + 4 T + p T^{2}
61 1+6T+pT2 1 + 6 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1−8T+pT2 1 - 8 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1+14T+pT2 1 + 14 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−8.090753670292665678650630726104, −7.19318728875240680365809782499, −6.56603709386798265186780548010, −6.12352424703812886192007841034, −4.80136783253141896105575045128, −4.34277649858975340857680013722, −3.35934720941182887444251376190, −2.61752725197143501613702660246, −1.32657258305720663270738938186, 0, 1.32657258305720663270738938186, 2.61752725197143501613702660246, 3.35934720941182887444251376190, 4.34277649858975340857680013722, 4.80136783253141896105575045128, 6.12352424703812886192007841034, 6.56603709386798265186780548010, 7.19318728875240680365809782499, 8.090753670292665678650630726104

Graph of the ZZ-function along the critical line