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Properties

Label 2-2240-1.1-c3-0-38
Degree 22
Conductor 22402240
Sign 11
Analytic cond. 132.164132.164
Root an. cond. 11.496211.4962
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.61·3-s − 5·5-s + 7·7-s − 13.9·9-s − 68.4·11-s + 91.8·13-s − 18.0·15-s + 77.1·17-s − 55.5·19-s + 25.2·21-s − 97.8·23-s + 25·25-s − 147.·27-s + 20.3·29-s − 29.7·31-s − 247.·33-s − 35·35-s + 133.·37-s + 331.·39-s + 99.8·41-s + 298.·43-s + 69.6·45-s + 353.·47-s + 49·49-s + 278.·51-s − 501.·53-s + 342.·55-s + ⋯
L(s)  = 1  + 0.695·3-s − 0.447·5-s + 0.377·7-s − 0.516·9-s − 1.87·11-s + 1.95·13-s − 0.311·15-s + 1.10·17-s − 0.670·19-s + 0.262·21-s − 0.887·23-s + 0.200·25-s − 1.05·27-s + 0.130·29-s − 0.172·31-s − 1.30·33-s − 0.169·35-s + 0.591·37-s + 1.36·39-s + 0.380·41-s + 1.05·43-s + 0.230·45-s + 1.09·47-s + 0.142·49-s + 0.765·51-s − 1.29·53-s + 0.838·55-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: 11
Analytic conductor: 132.164132.164
Root analytic conductor: 11.496211.4962
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2240, ( :3/2), 1)(2,\ 2240,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.2163365212.216336521
L(12)L(\frac12) ≈\approx 2.2163365212.216336521
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+5T 1 + 5T
7 1−7T 1 - 7T
good3 1−3.61T+27T2 1 - 3.61T + 27T^{2}
11 1+68.4T+1.33e3T2 1 + 68.4T + 1.33e3T^{2}
13 1−91.8T+2.19e3T2 1 - 91.8T + 2.19e3T^{2}
17 1−77.1T+4.91e3T2 1 - 77.1T + 4.91e3T^{2}
19 1+55.5T+6.85e3T2 1 + 55.5T + 6.85e3T^{2}
23 1+97.8T+1.21e4T2 1 + 97.8T + 1.21e4T^{2}
29 1−20.3T+2.43e4T2 1 - 20.3T + 2.43e4T^{2}
31 1+29.7T+2.97e4T2 1 + 29.7T + 2.97e4T^{2}
37 1−133.T+5.06e4T2 1 - 133.T + 5.06e4T^{2}
41 1−99.8T+6.89e4T2 1 - 99.8T + 6.89e4T^{2}
43 1−298.T+7.95e4T2 1 - 298.T + 7.95e4T^{2}
47 1−353.T+1.03e5T2 1 - 353.T + 1.03e5T^{2}
53 1+501.T+1.48e5T2 1 + 501.T + 1.48e5T^{2}
59 1+638.T+2.05e5T2 1 + 638.T + 2.05e5T^{2}
61 1+165.T+2.26e5T2 1 + 165.T + 2.26e5T^{2}
67 1+480.T+3.00e5T2 1 + 480.T + 3.00e5T^{2}
71 1−1.03e3T+3.57e5T2 1 - 1.03e3T + 3.57e5T^{2}
73 1+28.8T+3.89e5T2 1 + 28.8T + 3.89e5T^{2}
79 1+367.T+4.93e5T2 1 + 367.T + 4.93e5T^{2}
83 1+121.T+5.71e5T2 1 + 121.T + 5.71e5T^{2}
89 1−1.54e3T+7.04e5T2 1 - 1.54e3T + 7.04e5T^{2}
97 1+65.1T+9.12e5T2 1 + 65.1T + 9.12e5T^{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

−8.507667473286162476949494898684, −7.894726291570960918054916341539, −7.64897761849668926936454064235, −6.10202227656284747087177086846, −5.69036941496381106886142960759, −4.57312407080089825837674861717, −3.61215436915000769378072472470, −2.93192825938238716012178908439, −1.95375325753614988450958731553, −0.63276309394581083436649328531, 0.63276309394581083436649328531, 1.95375325753614988450958731553, 2.93192825938238716012178908439, 3.61215436915000769378072472470, 4.57312407080089825837674861717, 5.69036941496381106886142960759, 6.10202227656284747087177086846, 7.64897761849668926936454064235, 7.894726291570960918054916341539, 8.507667473286162476949494898684

Graph of the ZZ-function along the critical line