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Properties

Label 2-3744-1.1-c1-0-38
Degree 22
Conductor 37443744
Sign −1-1
Analytic cond. 29.895929.8959
Root an. cond. 5.467725.46772
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 − 3·7-s + 2·11-s + 13-s + 3·17-s − 2·19-s + 4·23-s − 4·25-s − 2·29-s − 4·31-s + 3·35-s + 5·37-s + 12·41-s − 7·43-s − 9·47-s + 2·49-s − 4·53-s − 2·55-s + 6·59-s − 4·61-s − 65-s + 10·67-s − 15·71-s − 2·73-s − 6·77-s + 8·79-s − 4·83-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.13·7-s + 0.603·11-s + 0.277·13-s + 0.727·17-s − 0.458·19-s + 0.834·23-s − 4/5·25-s − 0.371·29-s − 0.718·31-s + 0.507·35-s + 0.821·37-s + 1.87·41-s − 1.06·43-s − 1.31·47-s + 2/7·49-s − 0.549·53-s − 0.269·55-s + 0.781·59-s − 0.512·61-s − 0.124·65-s + 1.22·67-s − 1.78·71-s − 0.234·73-s − 0.683·77-s + 0.900·79-s − 0.439·83-s + ⋯

Functional equation

Λ(s)=(3744s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3744s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37443744    =    25⋅32⋅132^{5} \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 29.895929.8959
Root analytic conductor: 5.467725.46772
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3744, ( :1/2), −1)(2,\ 3744,\ (\ :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
13 1−T 1 - T
good5 1+T+pT2 1 + T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
11 1−2T+pT2 1 - 2 T + p T^{2}
17 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+2T+pT2 1 + 2 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−5T+pT2 1 - 5 T + p T^{2}
41 1−12T+pT2 1 - 12 T + p T^{2}
43 1+7T+pT2 1 + 7 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1+4T+pT2 1 + 4 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1−10T+pT2 1 - 10 T + p T^{2}
71 1+15T+pT2 1 + 15 T + p T^{2}
73 1+2T+pT2 1 + 2 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1+4T+pT2 1 + 4 T + p T^{2}
89 1+2T+pT2 1 + 2 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

−8.069287068407326044308270083618, −7.43349489084068026273091343648, −6.58077620798120174116715076060, −6.08062567625954929148194792858, −5.16574128381926105856746207319, −4.07091054811134105912777170722, −3.54020682240368361077395499750, −2.67039823935118854544464566270, −1.33035396953510359861485977690, 0, 1.33035396953510359861485977690, 2.67039823935118854544464566270, 3.54020682240368361077395499750, 4.07091054811134105912777170722, 5.16574128381926105856746207319, 6.08062567625954929148194792858, 6.58077620798120174116715076060, 7.43349489084068026273091343648, 8.069287068407326044308270083618

Graph of the ZZ-function along the critical line