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Properties

Label 2-3276-1.1-c1-0-10
Degree 22
Conductor 32763276
Sign 11
Analytic cond. 26.158926.1589
Root an. cond. 5.114585.11458
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  − 5-s + 7-s + 6·11-s − 13-s + 8·17-s + 3·19-s − 9·23-s − 4·25-s + 9·29-s − 31-s − 35-s − 8·37-s − 2·41-s + 9·43-s + 3·47-s + 49-s + 5·53-s − 6·55-s − 8·59-s − 10·61-s + 65-s − 10·67-s + 12·71-s + 3·73-s + 6·77-s − 7·79-s + 15·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 1.80·11-s − 0.277·13-s + 1.94·17-s + 0.688·19-s − 1.87·23-s − 4/5·25-s + 1.67·29-s − 0.179·31-s − 0.169·35-s − 1.31·37-s − 0.312·41-s + 1.37·43-s + 0.437·47-s + 1/7·49-s + 0.686·53-s − 0.809·55-s − 1.04·59-s − 1.28·61-s + 0.124·65-s − 1.22·67-s + 1.42·71-s + 0.351·73-s + 0.683·77-s − 0.787·79-s + 1.64·83-s + ⋯

Functional equation

Λ(s)=(3276s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3276s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32763276    =    22⋅32⋅7⋅132^{2} \cdot 3^{2} \cdot 7 \cdot 13
Sign: 11
Analytic conductor: 26.158926.1589
Root analytic conductor: 5.114585.11458
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3276, ( :1/2), 1)(2,\ 3276,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1020435452.102043545
L(12)L(\frac12) ≈\approx 2.1020435452.102043545
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−T 1 - T
13 1+T 1 + T
good5 1+T+pT2 1 + T + p T^{2}
11 1−6T+pT2 1 - 6 T + p T^{2}
17 1−8T+pT2 1 - 8 T + p T^{2}
19 1−3T+pT2 1 - 3 T + p T^{2}
23 1+9T+pT2 1 + 9 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1+T+pT2 1 + T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−9T+pT2 1 - 9 T + p T^{2}
47 1−3T+pT2 1 - 3 T + p T^{2}
53 1−5T+pT2 1 - 5 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+10T+pT2 1 + 10 T + p T^{2}
67 1+10T+pT2 1 + 10 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−3T+pT2 1 - 3 T + p T^{2}
79 1+7T+pT2 1 + 7 T + p T^{2}
83 1−15T+pT2 1 - 15 T + p T^{2}
89 1−5T+pT2 1 - 5 T + p T^{2}
97 1−15T+pT2 1 - 15 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

−8.588943294913916144303770226519, −7.81017917770960962493477778363, −7.34720187422893546120672011284, −6.29947170772173464283060055039, −5.73838646791182021882632100284, −4.69034388954423580716740084793, −3.88102117197123216567783894059, −3.28850193522786672972611735122, −1.87027540489607850169666877892, −0.927402921974082899230984722556, 0.927402921974082899230984722556, 1.87027540489607850169666877892, 3.28850193522786672972611735122, 3.88102117197123216567783894059, 4.69034388954423580716740084793, 5.73838646791182021882632100284, 6.29947170772173464283060055039, 7.34720187422893546120672011284, 7.81017917770960962493477778363, 8.588943294913916144303770226519

Graph of the ZZ-function along the critical line