↑

Properties

Label 2-538-1.1-c1-0-9
Degree 22
Conductor 538538
Sign 11
Analytic cond. 4.295954.29595
Root an. cond. 2.072662.07266
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3.04·3-s + 4-s + 1.48·5-s − 3.04·6-s + 0.344·7-s − 8-s + 6.29·9-s − 1.48·10-s − 0.905·11-s + 3.04·12-s − 2.24·13-s − 0.344·14-s + 4.53·15-s + 16-s + 2.58·17-s − 6.29·18-s − 0.688·19-s + 1.48·20-s + 1.04·21-s + 0.905·22-s + 1.46·23-s − 3.04·24-s − 2.78·25-s + 2.24·26-s + 10.0·27-s + 0.344·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.76·3-s + 0.5·4-s + 0.665·5-s − 1.24·6-s + 0.130·7-s − 0.353·8-s + 2.09·9-s − 0.470·10-s − 0.273·11-s + 0.880·12-s − 0.623·13-s − 0.0919·14-s + 1.17·15-s + 0.250·16-s + 0.627·17-s − 1.48·18-s − 0.157·19-s + 0.332·20-s + 0.229·21-s + 0.193·22-s + 0.304·23-s − 0.622·24-s − 0.557·25-s + 0.441·26-s + 1.93·27-s + 0.0650·28-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 11
Analytic conductor: 4.295954.29595
Root analytic conductor: 2.072662.07266
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 538, ( :1/2), 1)(2,\ 538,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.0675396812.067539681
L(12)L(\frac12) ≈\approx 2.0675396812.067539681
L(32)L(\frac{3}{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+T 1 + T
269 1−T 1 - T
good3 1−3.04T+3T2 1 - 3.04T + 3T^{2}
5 1−1.48T+5T2 1 - 1.48T + 5T^{2}
7 1−0.344T+7T2 1 - 0.344T + 7T^{2}
11 1+0.905T+11T2 1 + 0.905T + 11T^{2}
13 1+2.24T+13T2 1 + 2.24T + 13T^{2}
17 1−2.58T+17T2 1 - 2.58T + 17T^{2}
19 1+0.688T+19T2 1 + 0.688T + 19T^{2}
23 1−1.46T+23T2 1 - 1.46T + 23T^{2}
29 1+4.24T+29T2 1 + 4.24T + 29T^{2}
31 1−1.92T+31T2 1 - 1.92T + 31T^{2}
37 1+1.84T+37T2 1 + 1.84T + 37T^{2}
41 1−1.21T+41T2 1 - 1.21T + 41T^{2}
43 1+6.59T+43T2 1 + 6.59T + 43T^{2}
47 1+6.42T+47T2 1 + 6.42T + 47T^{2}
53 1−2.87T+53T2 1 - 2.87T + 53T^{2}
59 1−11.0T+59T2 1 - 11.0T + 59T^{2}
61 1+1.75T+61T2 1 + 1.75T + 61T^{2}
67 1−4.34T+67T2 1 - 4.34T + 67T^{2}
71 1−5.29T+71T2 1 - 5.29T + 71T^{2}
73 1−5.90T+73T2 1 - 5.90T + 73T^{2}
79 1+15.1T+79T2 1 + 15.1T + 79T^{2}
83 1−15.7T+83T2 1 - 15.7T + 83T^{2}
89 1+1.61T+89T2 1 + 1.61T + 89T^{2}
97 1−14.3T+97T2 1 - 14.3T + 97T^{2}
show more
show less
   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

−10.27294859026301331102160398984, −9.772291867283740368337284756728, −9.096558459979297704909355160644, −8.200609362081373821187882845746, −7.62246017138901322808141009128, −6.63178167518341311723693321948, −5.18577238747919645306811178745, −3.68149435561549818431328879875, −2.61259424255856168354097745297, −1.71212161812236731954651064958, 1.71212161812236731954651064958, 2.61259424255856168354097745297, 3.68149435561549818431328879875, 5.18577238747919645306811178745, 6.63178167518341311723693321948, 7.62246017138901322808141009128, 8.200609362081373821187882845746, 9.096558459979297704909355160644, 9.772291867283740368337284756728, 10.27294859026301331102160398984

Graph of the ZZ-function along the critical line