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Properties

Label 2-2112-1.1-c3-0-31
Degree 22
Conductor 21122112
Sign 11
Analytic cond. 124.612124.612
Root an. cond. 11.162911.1629
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  − 3·3-s + 3.48·5-s − 4.74·7-s + 9·9-s − 11·11-s + 15.0·13-s − 10.4·15-s + 73.1·17-s + 78.7·19-s + 14.2·21-s + 112·23-s − 112.·25-s − 27·27-s − 243.·29-s + 278.·31-s + 33·33-s − 16.5·35-s − 102.·37-s − 45.0·39-s − 241.·41-s + 280.·43-s + 31.4·45-s − 169.·47-s − 320.·49-s − 219.·51-s + 409.·53-s − 38.3·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.312·5-s − 0.256·7-s + 0.333·9-s − 0.301·11-s + 0.320·13-s − 0.180·15-s + 1.04·17-s + 0.950·19-s + 0.147·21-s + 1.01·23-s − 0.902·25-s − 0.192·27-s − 1.55·29-s + 1.61·31-s + 0.174·33-s − 0.0799·35-s − 0.454·37-s − 0.185·39-s − 0.918·41-s + 0.993·43-s + 0.104·45-s − 0.527·47-s − 0.934·49-s − 0.602·51-s + 1.06·53-s − 0.0940·55-s + ⋯

Functional equation

Λ(s)=(2112s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2112s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21122112    =    26⋅3⋅112^{6} \cdot 3 \cdot 11
Sign: 11
Analytic conductor: 124.612124.612
Root analytic conductor: 11.162911.1629
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2112, ( :3/2), 1)(2,\ 2112,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.8307273541.830727354
L(12)L(\frac12) ≈\approx 1.8307273541.830727354
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+3T 1 + 3T
11 1+11T 1 + 11T
good5 1−3.48T+125T2 1 - 3.48T + 125T^{2}
7 1+4.74T+343T2 1 + 4.74T + 343T^{2}
13 1−15.0T+2.19e3T2 1 - 15.0T + 2.19e3T^{2}
17 1−73.1T+4.91e3T2 1 - 73.1T + 4.91e3T^{2}
19 1−78.7T+6.85e3T2 1 - 78.7T + 6.85e3T^{2}
23 1−112T+1.21e4T2 1 - 112T + 1.21e4T^{2}
29 1+243.T+2.43e4T2 1 + 243.T + 2.43e4T^{2}
31 1−278.T+2.97e4T2 1 - 278.T + 2.97e4T^{2}
37 1+102.T+5.06e4T2 1 + 102.T + 5.06e4T^{2}
41 1+241.T+6.89e4T2 1 + 241.T + 6.89e4T^{2}
43 1−280.T+7.95e4T2 1 - 280.T + 7.95e4T^{2}
47 1+169.T+1.03e5T2 1 + 169.T + 1.03e5T^{2}
53 1−409.T+1.48e5T2 1 - 409.T + 1.48e5T^{2}
59 1+196T+2.05e5T2 1 + 196T + 2.05e5T^{2}
61 1−701.T+2.26e5T2 1 - 701.T + 2.26e5T^{2}
67 1+900.T+3.00e5T2 1 + 900.T + 3.00e5T^{2}
71 1−756.T+3.57e5T2 1 - 756.T + 3.57e5T^{2}
73 1+1.01e3T+3.89e5T2 1 + 1.01e3T + 3.89e5T^{2}
79 1+327.T+4.93e5T2 1 + 327.T + 4.93e5T^{2}
83 1−756.T+5.71e5T2 1 - 756.T + 5.71e5T^{2}
89 1−508.T+7.04e5T2 1 - 508.T + 7.04e5T^{2}
97 1−614.T+9.12e5T2 1 - 614.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

−8.815810639442874799172705007753, −7.82227227478686751033263048979, −7.20899282042960774437768775724, −6.26021005530900535605219196833, −5.57833888984777959040823979141, −4.94777553986942266350621156727, −3.77420062097868305644775264830, −2.94376281174833214094834403519, −1.65220505299215808183299885422, −0.65723672139089052886033253474, 0.65723672139089052886033253474, 1.65220505299215808183299885422, 2.94376281174833214094834403519, 3.77420062097868305644775264830, 4.94777553986942266350621156727, 5.57833888984777959040823979141, 6.26021005530900535605219196833, 7.20899282042960774437768775724, 7.82227227478686751033263048979, 8.815810639442874799172705007753

Graph of the ZZ-function along the critical line