Properties

Label 2-538-1.1-c7-0-18
Degree 22
Conductor 538538
Sign 11
Analytic cond. 168.063168.063
Root an. cond. 12.963912.9639
Motivic weight 77
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 24.3·3-s + 64·4-s − 116.·5-s + 194.·6-s + 1.26e3·7-s − 512·8-s − 1.59e3·9-s + 933.·10-s + 3.68e3·11-s − 1.55e3·12-s − 1.45e4·13-s − 1.01e4·14-s + 2.84e3·15-s + 4.09e3·16-s − 2.39e4·17-s + 1.27e4·18-s − 1.48e3·19-s − 7.46e3·20-s − 3.08e4·21-s − 2.94e4·22-s + 3.46e4·23-s + 1.24e4·24-s − 6.45e4·25-s + 1.16e5·26-s + 9.21e4·27-s + 8.10e4·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.520·3-s + 0.5·4-s − 0.417·5-s + 0.368·6-s + 1.39·7-s − 0.353·8-s − 0.728·9-s + 0.295·10-s + 0.834·11-s − 0.260·12-s − 1.84·13-s − 0.987·14-s + 0.217·15-s + 0.250·16-s − 1.18·17-s + 0.515·18-s − 0.0495·19-s − 0.208·20-s − 0.727·21-s − 0.590·22-s + 0.593·23-s + 0.184·24-s − 0.825·25-s + 1.30·26-s + 0.900·27-s + 0.698·28-s + ⋯

Functional equation

Λ(s)=(538s/2ΓC(s)L(s)=(Λ(8s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(538s/2ΓC(s+7/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    22692 \cdot 269
Sign: 11
Analytic conductor: 168.063168.063
Root analytic conductor: 12.963912.9639
Motivic weight: 77
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 538, ( :7/2), 1)(2,\ 538,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) \approx 0.65655931860.6565593186
L(12)L(\frac12) \approx 0.65655931860.6565593186
L(92)L(\frac{9}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+8T 1 + 8T
269 1+1.94e7T 1 + 1.94e7T
good3 1+24.3T+2.18e3T2 1 + 24.3T + 2.18e3T^{2}
5 1+116.T+7.81e4T2 1 + 116.T + 7.81e4T^{2}
7 11.26e3T+8.23e5T2 1 - 1.26e3T + 8.23e5T^{2}
11 13.68e3T+1.94e7T2 1 - 3.68e3T + 1.94e7T^{2}
13 1+1.45e4T+6.27e7T2 1 + 1.45e4T + 6.27e7T^{2}
17 1+2.39e4T+4.10e8T2 1 + 2.39e4T + 4.10e8T^{2}
19 1+1.48e3T+8.93e8T2 1 + 1.48e3T + 8.93e8T^{2}
23 13.46e4T+3.40e9T2 1 - 3.46e4T + 3.40e9T^{2}
29 1+4.79e4T+1.72e10T2 1 + 4.79e4T + 1.72e10T^{2}
31 1+2.58e5T+2.75e10T2 1 + 2.58e5T + 2.75e10T^{2}
37 13.73e5T+9.49e10T2 1 - 3.73e5T + 9.49e10T^{2}
41 13.29e5T+1.94e11T2 1 - 3.29e5T + 1.94e11T^{2}
43 15.36e5T+2.71e11T2 1 - 5.36e5T + 2.71e11T^{2}
47 16.23e4T+5.06e11T2 1 - 6.23e4T + 5.06e11T^{2}
53 1+1.06e6T+1.17e12T2 1 + 1.06e6T + 1.17e12T^{2}
59 1+2.35e6T+2.48e12T2 1 + 2.35e6T + 2.48e12T^{2}
61 13.98e5T+3.14e12T2 1 - 3.98e5T + 3.14e12T^{2}
67 1+3.21e6T+6.06e12T2 1 + 3.21e6T + 6.06e12T^{2}
71 13.83e6T+9.09e12T2 1 - 3.83e6T + 9.09e12T^{2}
73 1+4.28e6T+1.10e13T2 1 + 4.28e6T + 1.10e13T^{2}
79 15.12e6T+1.92e13T2 1 - 5.12e6T + 1.92e13T^{2}
83 1+2.90e6T+2.71e13T2 1 + 2.90e6T + 2.71e13T^{2}
89 1+1.03e7T+4.42e13T2 1 + 1.03e7T + 4.42e13T^{2}
97 11.17e7T+8.07e13T2 1 - 1.17e7T + 8.07e13T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.493238120886446332516831831805, −8.888306376044114532218024080843, −7.80920513926631521106519575728, −7.28666583864275565658552493812, −6.10483342011451388547862282900, −5.05707603792514387455766598271, −4.24398143543010063135230630815, −2.61318125640387370724291688577, −1.68147615998146276550705001117, −0.39645870772114069840500024429, 0.39645870772114069840500024429, 1.68147615998146276550705001117, 2.61318125640387370724291688577, 4.24398143543010063135230630815, 5.05707603792514387455766598271, 6.10483342011451388547862282900, 7.28666583864275565658552493812, 7.80920513926631521106519575728, 8.888306376044114532218024080843, 9.493238120886446332516831831805

Graph of the ZZ-function along the critical line