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Properties

Label 2-2240-1.1-c3-0-120
Degree 22
Conductor 22402240
Sign −1-1
Analytic cond. 132.164132.164
Root an. cond. 11.496211.4962
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s − 5·5-s + 7·7-s − 11·9-s − 20·11-s + 10·13-s − 20·15-s − 14·17-s − 12·19-s + 28·21-s + 104·23-s + 25·25-s − 152·27-s + 122·29-s + 224·31-s − 80·33-s − 35·35-s − 158·37-s + 40·39-s + 378·41-s − 404·43-s + 55·45-s + 112·47-s + 49·49-s − 56·51-s − 270·53-s + 100·55-s + ⋯
L(s)  = 1  + 0.769·3-s − 0.447·5-s + 0.377·7-s − 0.407·9-s − 0.548·11-s + 0.213·13-s − 0.344·15-s − 0.199·17-s − 0.144·19-s + 0.290·21-s + 0.942·23-s + 1/5·25-s − 1.08·27-s + 0.781·29-s + 1.29·31-s − 0.422·33-s − 0.169·35-s − 0.702·37-s + 0.164·39-s + 1.43·41-s − 1.43·43-s + 0.182·45-s + 0.347·47-s + 1/7·49-s − 0.153·51-s − 0.699·53-s + 0.245·55-s + ⋯

Functional equation

Λ(s)=(2240s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(2240s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22402240    =    26⋅5⋅72^{6} \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 132.164132.164
Root analytic conductor: 11.496211.4962
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2240, ( :3/2), −1)(2,\ 2240,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
5 1+pT 1 + p T
7 1−pT 1 - p T
good3 1−4T+p3T2 1 - 4 T + p^{3} T^{2}
11 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
13 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
17 1+14T+p3T2 1 + 14 T + p^{3} T^{2}
19 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
23 1−104T+p3T2 1 - 104 T + p^{3} T^{2}
29 1−122T+p3T2 1 - 122 T + p^{3} T^{2}
31 1−224T+p3T2 1 - 224 T + p^{3} T^{2}
37 1+158T+p3T2 1 + 158 T + p^{3} T^{2}
41 1−378T+p3T2 1 - 378 T + p^{3} T^{2}
43 1+404T+p3T2 1 + 404 T + p^{3} T^{2}
47 1−112T+p3T2 1 - 112 T + p^{3} T^{2}
53 1+270T+p3T2 1 + 270 T + p^{3} T^{2}
59 1+324T+p3T2 1 + 324 T + p^{3} T^{2}
61 1−186T+p3T2 1 - 186 T + p^{3} T^{2}
67 1+156T+p3T2 1 + 156 T + p^{3} T^{2}
71 1+360T+p3T2 1 + 360 T + p^{3} T^{2}
73 1+102T+p3T2 1 + 102 T + p^{3} T^{2}
79 1+912T+p3T2 1 + 912 T + p^{3} T^{2}
83 1+1068T+p3T2 1 + 1068 T + p^{3} T^{2}
89 1+1590T+p3T2 1 + 1590 T + p^{3} T^{2}
97 1−866T+p3T2 1 - 866 T + p^{3} 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

−8.401396749705539678280364072113, −7.71072616451701854255082284213, −6.90420882731457873032848783538, −5.93265427519409831309417099220, −4.98390944568426141396845394694, −4.20903758782323131532550554053, −3.12473531118455861327321136478, −2.57931620538888695612279615245, −1.31029344486531402605357430268, 0, 1.31029344486531402605357430268, 2.57931620538888695612279615245, 3.12473531118455861327321136478, 4.20903758782323131532550554053, 4.98390944568426141396845394694, 5.93265427519409831309417099220, 6.90420882731457873032848783538, 7.71072616451701854255082284213, 8.401396749705539678280364072113

Graph of the ZZ-function along the critical line