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Properties

Label 2-1152-1.1-c3-0-23
Degree 22
Conductor 11521152
Sign 11
Analytic cond. 67.970267.9702
Root an. cond. 8.244408.24440
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  + 11.8·5-s − 9.85·7-s + 39.0·11-s + 91.5·13-s + 37.1·17-s + 46.4·19-s − 120.·23-s + 15.5·25-s + 27.2·29-s + 81.1·31-s − 116.·35-s + 10.9·37-s + 205.·41-s − 115.·43-s − 312.·47-s − 245.·49-s − 90.9·53-s + 463.·55-s − 550.·59-s + 630.·61-s + 1.08e3·65-s + 661.·67-s + 494.·71-s + 566.·73-s − 385.·77-s − 49.4·79-s + 564.·83-s + ⋯
L(s)  = 1  + 1.06·5-s − 0.532·7-s + 1.07·11-s + 1.95·13-s + 0.529·17-s + 0.561·19-s − 1.09·23-s + 0.124·25-s + 0.174·29-s + 0.470·31-s − 0.564·35-s + 0.0488·37-s + 0.782·41-s − 0.409·43-s − 0.970·47-s − 0.716·49-s − 0.235·53-s + 1.13·55-s − 1.21·59-s + 1.32·61-s + 2.07·65-s + 1.20·67-s + 0.827·71-s + 0.907·73-s − 0.569·77-s − 0.0703·79-s + 0.745·83-s + ⋯

Functional equation

Λ(s)=(1152s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1152s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11521152    =    27⋅322^{7} \cdot 3^{2}
Sign: 11
Analytic conductor: 67.970267.9702
Root analytic conductor: 8.244408.24440
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1152, ( :3/2), 1)(2,\ 1152,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.0821156553.082115655
L(12)L(\frac12) ≈\approx 3.0821156553.082115655
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
3 1 1
good5 1−11.8T+125T2 1 - 11.8T + 125T^{2}
7 1+9.85T+343T2 1 + 9.85T + 343T^{2}
11 1−39.0T+1.33e3T2 1 - 39.0T + 1.33e3T^{2}
13 1−91.5T+2.19e3T2 1 - 91.5T + 2.19e3T^{2}
17 1−37.1T+4.91e3T2 1 - 37.1T + 4.91e3T^{2}
19 1−46.4T+6.85e3T2 1 - 46.4T + 6.85e3T^{2}
23 1+120.T+1.21e4T2 1 + 120.T + 1.21e4T^{2}
29 1−27.2T+2.43e4T2 1 - 27.2T + 2.43e4T^{2}
31 1−81.1T+2.97e4T2 1 - 81.1T + 2.97e4T^{2}
37 1−10.9T+5.06e4T2 1 - 10.9T + 5.06e4T^{2}
41 1−205.T+6.89e4T2 1 - 205.T + 6.89e4T^{2}
43 1+115.T+7.95e4T2 1 + 115.T + 7.95e4T^{2}
47 1+312.T+1.03e5T2 1 + 312.T + 1.03e5T^{2}
53 1+90.9T+1.48e5T2 1 + 90.9T + 1.48e5T^{2}
59 1+550.T+2.05e5T2 1 + 550.T + 2.05e5T^{2}
61 1−630.T+2.26e5T2 1 - 630.T + 2.26e5T^{2}
67 1−661.T+3.00e5T2 1 - 661.T + 3.00e5T^{2}
71 1−494.T+3.57e5T2 1 - 494.T + 3.57e5T^{2}
73 1−566.T+3.89e5T2 1 - 566.T + 3.89e5T^{2}
79 1+49.4T+4.93e5T2 1 + 49.4T + 4.93e5T^{2}
83 1−564.T+5.71e5T2 1 - 564.T + 5.71e5T^{2}
89 1+1.08e3T+7.04e5T2 1 + 1.08e3T + 7.04e5T^{2}
97 1−464.T+9.12e5T2 1 - 464.T + 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

−9.573246124929127709405737165841, −8.711883612088204410309204229018, −7.900795239583206496533982475286, −6.45662512772631035007470417003, −6.29489710607757447726469706472, −5.37733307028289133126446210314, −4.00351961040418126977313039840, −3.27090056278619453963928088953, −1.86172203544796204926168754380, −0.983132755291107895015842329883, 0.983132755291107895015842329883, 1.86172203544796204926168754380, 3.27090056278619453963928088953, 4.00351961040418126977313039840, 5.37733307028289133126446210314, 6.29489710607757447726469706472, 6.45662512772631035007470417003, 7.900795239583206496533982475286, 8.711883612088204410309204229018, 9.573246124929127709405737165841

Graph of the ZZ-function along the critical line