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Properties

Label 2-1472-1.1-c3-0-48
Degree 22
Conductor 14721472
Sign 11
Analytic cond. 86.850886.8508
Root an. cond. 9.319379.31937
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  − 7.72·3-s − 18.6·5-s + 28.1·7-s + 32.7·9-s + 44.4·11-s + 67.2·13-s + 143.·15-s − 21.2·17-s + 151.·19-s − 217.·21-s − 23·23-s + 221.·25-s − 44.0·27-s + 164.·29-s + 120.·31-s − 343.·33-s − 524.·35-s + 188.·37-s − 519.·39-s − 59.5·41-s − 432.·43-s − 608.·45-s + 555.·47-s + 451.·49-s + 164.·51-s + 106.·53-s − 827.·55-s + ⋯
L(s)  = 1  − 1.48·3-s − 1.66·5-s + 1.52·7-s + 1.21·9-s + 1.21·11-s + 1.43·13-s + 2.47·15-s − 0.303·17-s + 1.82·19-s − 2.26·21-s − 0.208·23-s + 1.77·25-s − 0.314·27-s + 1.05·29-s + 0.700·31-s − 1.81·33-s − 2.53·35-s + 0.839·37-s − 2.13·39-s − 0.226·41-s − 1.53·43-s − 2.01·45-s + 1.72·47-s + 1.31·49-s + 0.450·51-s + 0.276·53-s − 2.02·55-s + ⋯

Functional equation

Λ(s)=(1472s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1472s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14721472    =    26⋅232^{6} \cdot 23
Sign: 11
Analytic conductor: 86.850886.8508
Root analytic conductor: 9.319379.31937
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1472, ( :3/2), 1)(2,\ 1472,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.4761782191.476178219
L(12)L(\frac12) ≈\approx 1.4761782191.476178219
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
23 1+23T 1 + 23T
good3 1+7.72T+27T2 1 + 7.72T + 27T^{2}
5 1+18.6T+125T2 1 + 18.6T + 125T^{2}
7 1−28.1T+343T2 1 - 28.1T + 343T^{2}
11 1−44.4T+1.33e3T2 1 - 44.4T + 1.33e3T^{2}
13 1−67.2T+2.19e3T2 1 - 67.2T + 2.19e3T^{2}
17 1+21.2T+4.91e3T2 1 + 21.2T + 4.91e3T^{2}
19 1−151.T+6.85e3T2 1 - 151.T + 6.85e3T^{2}
29 1−164.T+2.43e4T2 1 - 164.T + 2.43e4T^{2}
31 1−120.T+2.97e4T2 1 - 120.T + 2.97e4T^{2}
37 1−188.T+5.06e4T2 1 - 188.T + 5.06e4T^{2}
41 1+59.5T+6.89e4T2 1 + 59.5T + 6.89e4T^{2}
43 1+432.T+7.95e4T2 1 + 432.T + 7.95e4T^{2}
47 1−555.T+1.03e5T2 1 - 555.T + 1.03e5T^{2}
53 1−106.T+1.48e5T2 1 - 106.T + 1.48e5T^{2}
59 1+571.T+2.05e5T2 1 + 571.T + 2.05e5T^{2}
61 1−37.4T+2.26e5T2 1 - 37.4T + 2.26e5T^{2}
67 1+94.5T+3.00e5T2 1 + 94.5T + 3.00e5T^{2}
71 1+6.85T+3.57e5T2 1 + 6.85T + 3.57e5T^{2}
73 1−194.T+3.89e5T2 1 - 194.T + 3.89e5T^{2}
79 1+12.6T+4.93e5T2 1 + 12.6T + 4.93e5T^{2}
83 1+194.T+5.71e5T2 1 + 194.T + 5.71e5T^{2}
89 1+682.T+7.04e5T2 1 + 682.T + 7.04e5T^{2}
97 1−1.12e3T+9.12e5T2 1 - 1.12e3T + 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

−8.908716730556660381906235141878, −8.237425085007902047080101327433, −7.49958533632258940379596541152, −6.68529149947993891720702869995, −5.82203570554423947594225853644, −4.81091255969224530019384654138, −4.31959279849179437354974343203, −3.40605705360710064514985473825, −1.27723077123145478346334681322, −0.798678382865003062382215297000, 0.798678382865003062382215297000, 1.27723077123145478346334681322, 3.40605705360710064514985473825, 4.31959279849179437354974343203, 4.81091255969224530019384654138, 5.82203570554423947594225853644, 6.68529149947993891720702869995, 7.49958533632258940379596541152, 8.237425085007902047080101327433, 8.908716730556660381906235141878

Graph of the ZZ-function along the critical line