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Properties

Label 2-10-1.1-c11-0-1
Degree 22
Conductor 1010
Sign 11
Analytic cond. 7.683437.68343
Root an. cond. 2.771902.77190
Motivic weight 1111
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 32·2-s − 141.·3-s + 1.02e3·4-s + 3.12e3·5-s − 4.53e3·6-s + 8.55e4·7-s + 3.27e4·8-s − 1.57e5·9-s + 1.00e5·10-s + 7.67e5·11-s − 1.45e5·12-s + 2.20e5·13-s + 2.73e6·14-s − 4.42e5·15-s + 1.04e6·16-s − 9.30e5·17-s − 5.02e6·18-s − 1.77e7·19-s + 3.20e6·20-s − 1.21e7·21-s + 2.45e7·22-s − 3.99e7·23-s − 4.64e6·24-s + 9.76e6·25-s + 7.07e6·26-s + 4.73e7·27-s + 8.76e7·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.336·3-s + 0.5·4-s + 0.447·5-s − 0.238·6-s + 1.92·7-s + 0.353·8-s − 0.886·9-s + 0.316·10-s + 1.43·11-s − 0.168·12-s + 0.165·13-s + 1.36·14-s − 0.150·15-s + 0.250·16-s − 0.158·17-s − 0.626·18-s − 1.64·19-s + 0.223·20-s − 0.648·21-s + 1.01·22-s − 1.29·23-s − 0.119·24-s + 0.199·25-s + 0.116·26-s + 0.635·27-s + 0.962·28-s + ⋯

Functional equation

Λ(s)=(10s/2 ΓC(s) L(s)=( Λ(12−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}
Λ(s)=(10s/2 ΓC(s+11/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1010    =    2⋅52 \cdot 5
Sign: 11
Analytic conductor: 7.683437.68343
Root analytic conductor: 2.771902.77190
Motivic weight: 1111
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 10, ( :11/2), 1)(2,\ 10,\ (\ :11/2),\ 1)

Particular Values

L(6)L(6) ≈\approx 2.7117342192.711734219
L(12)L(\frac12) ≈\approx 2.7117342192.711734219
L(132)L(\frac{13}{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 1−32T 1 - 32T
5 1−3.12e3T 1 - 3.12e3T
good3 1+141.T+1.77e5T2 1 + 141.T + 1.77e5T^{2}
7 1−8.55e4T+1.97e9T2 1 - 8.55e4T + 1.97e9T^{2}
11 1−7.67e5T+2.85e11T2 1 - 7.67e5T + 2.85e11T^{2}
13 1−2.20e5T+1.79e12T2 1 - 2.20e5T + 1.79e12T^{2}
17 1+9.30e5T+3.42e13T2 1 + 9.30e5T + 3.42e13T^{2}
19 1+1.77e7T+1.16e14T2 1 + 1.77e7T + 1.16e14T^{2}
23 1+3.99e7T+9.52e14T2 1 + 3.99e7T + 9.52e14T^{2}
29 1−7.68e7T+1.22e16T2 1 - 7.68e7T + 1.22e16T^{2}
31 1+2.96e7T+2.54e16T2 1 + 2.96e7T + 2.54e16T^{2}
37 1−5.40e7T+1.77e17T2 1 - 5.40e7T + 1.77e17T^{2}
41 1−1.26e8T+5.50e17T2 1 - 1.26e8T + 5.50e17T^{2}
43 1+2.88e8T+9.29e17T2 1 + 2.88e8T + 9.29e17T^{2}
47 1+1.57e9T+2.47e18T2 1 + 1.57e9T + 2.47e18T^{2}
53 1+4.09e9T+9.26e18T2 1 + 4.09e9T + 9.26e18T^{2}
59 1−3.77e9T+3.01e19T2 1 - 3.77e9T + 3.01e19T^{2}
61 1+9.64e9T+4.35e19T2 1 + 9.64e9T + 4.35e19T^{2}
67 1+1.63e10T+1.22e20T2 1 + 1.63e10T + 1.22e20T^{2}
71 1−1.03e10T+2.31e20T2 1 - 1.03e10T + 2.31e20T^{2}
73 1−4.27e9T+3.13e20T2 1 - 4.27e9T + 3.13e20T^{2}
79 1+1.96e10T+7.47e20T2 1 + 1.96e10T + 7.47e20T^{2}
83 1+1.35e10T+1.28e21T2 1 + 1.35e10T + 1.28e21T^{2}
89 1−2.25e10T+2.77e21T2 1 - 2.25e10T + 2.77e21T^{2}
97 1+1.08e11T+7.15e21T2 1 + 1.08e11T + 7.15e21T^{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

−17.77736466117883358148823341509, −16.93640620326652268169203828394, −14.75357021979853328790989824818, −14.07432227422831341256009847652, −11.97272180346616251378113402968, −10.97230457851962835483947881259, −8.436532141699225614016171337797, −6.15015776511802460800897197579, −4.50776396697386038857954667873, −1.77322034821727295348141717469, 1.77322034821727295348141717469, 4.50776396697386038857954667873, 6.15015776511802460800897197579, 8.436532141699225614016171337797, 10.97230457851962835483947881259, 11.97272180346616251378113402968, 14.07432227422831341256009847652, 14.75357021979853328790989824818, 16.93640620326652268169203828394, 17.77736466117883358148823341509

Graph of the ZZ-function along the critical line