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Properties

Label 2-1872-1.1-c3-0-49
Degree 22
Conductor 18721872
Sign 11
Analytic cond. 110.451110.451
Root an. cond. 10.509510.5095
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  + 20.1·5-s + 19.6·7-s + 13.5·11-s + 13·13-s − 116.·17-s − 68.1·19-s + 122.·23-s + 280.·25-s − 204.·29-s + 194.·31-s + 396.·35-s − 142.·37-s + 175.·41-s + 219.·43-s + 236.·47-s + 45.0·49-s + 628.·53-s + 273.·55-s + 446.·59-s + 224.·61-s + 261.·65-s − 165.·67-s + 902.·71-s − 15.1·73-s + 266.·77-s − 670.·79-s + 1.04e3·83-s + ⋯
L(s)  = 1  + 1.80·5-s + 1.06·7-s + 0.371·11-s + 0.277·13-s − 1.66·17-s − 0.822·19-s + 1.10·23-s + 2.24·25-s − 1.30·29-s + 1.12·31-s + 1.91·35-s − 0.634·37-s + 0.666·41-s + 0.778·43-s + 0.733·47-s + 0.131·49-s + 1.62·53-s + 0.669·55-s + 0.984·59-s + 0.471·61-s + 0.499·65-s − 0.301·67-s + 1.50·71-s − 0.0242·73-s + 0.395·77-s − 0.954·79-s + 1.37·83-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 11
Analytic conductor: 110.451110.451
Root analytic conductor: 10.509510.5095
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1872, ( :3/2), 1)(2,\ 1872,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.9712219793.971221979
L(12)L(\frac12) ≈\approx 3.9712219793.971221979
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
13 1−13T 1 - 13T
good5 1−20.1T+125T2 1 - 20.1T + 125T^{2}
7 1−19.6T+343T2 1 - 19.6T + 343T^{2}
11 1−13.5T+1.33e3T2 1 - 13.5T + 1.33e3T^{2}
17 1+116.T+4.91e3T2 1 + 116.T + 4.91e3T^{2}
19 1+68.1T+6.85e3T2 1 + 68.1T + 6.85e3T^{2}
23 1−122.T+1.21e4T2 1 - 122.T + 1.21e4T^{2}
29 1+204.T+2.43e4T2 1 + 204.T + 2.43e4T^{2}
31 1−194.T+2.97e4T2 1 - 194.T + 2.97e4T^{2}
37 1+142.T+5.06e4T2 1 + 142.T + 5.06e4T^{2}
41 1−175.T+6.89e4T2 1 - 175.T + 6.89e4T^{2}
43 1−219.T+7.95e4T2 1 - 219.T + 7.95e4T^{2}
47 1−236.T+1.03e5T2 1 - 236.T + 1.03e5T^{2}
53 1−628.T+1.48e5T2 1 - 628.T + 1.48e5T^{2}
59 1−446.T+2.05e5T2 1 - 446.T + 2.05e5T^{2}
61 1−224.T+2.26e5T2 1 - 224.T + 2.26e5T^{2}
67 1+165.T+3.00e5T2 1 + 165.T + 3.00e5T^{2}
71 1−902.T+3.57e5T2 1 - 902.T + 3.57e5T^{2}
73 1+15.1T+3.89e5T2 1 + 15.1T + 3.89e5T^{2}
79 1+670.T+4.93e5T2 1 + 670.T + 4.93e5T^{2}
83 1−1.04e3T+5.71e5T2 1 - 1.04e3T + 5.71e5T^{2}
89 1−562.T+7.04e5T2 1 - 562.T + 7.04e5T^{2}
97 1−1.64e3T+9.12e5T2 1 - 1.64e3T + 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

−8.979612251648556713387301904354, −8.342495738358660732962068077466, −7.07314262565942452783780317368, −6.46761007700977991576663712401, −5.64407989849702682027840272393, −4.93482364942701458627533193608, −4.07033430036915002789681059487, −2.48043625507977346678838467219, −1.98055245855744336167724019081, −0.979320264978208541398548136926, 0.979320264978208541398548136926, 1.98055245855744336167724019081, 2.48043625507977346678838467219, 4.07033430036915002789681059487, 4.93482364942701458627533193608, 5.64407989849702682027840272393, 6.46761007700977991576663712401, 7.07314262565942452783780317368, 8.342495738358660732962068077466, 8.979612251648556713387301904354

Graph of the ZZ-function along the critical line