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Properties

Label 2-538-1.1-c5-0-69
Degree 22
Conductor 538538
Sign 11
Analytic cond. 86.286486.2864
Root an. cond. 9.289059.28905
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 26.7·3-s + 16·4-s + 50.5·5-s − 106.·6-s + 248.·7-s − 64·8-s + 472.·9-s − 202.·10-s + 32.9·11-s + 427.·12-s − 501.·13-s − 992.·14-s + 1.35e3·15-s + 256·16-s + 658.·17-s − 1.89e3·18-s − 1.91e3·19-s + 808.·20-s + 6.63e3·21-s − 131.·22-s + 3.25e3·23-s − 1.71e3·24-s − 568.·25-s + 2.00e3·26-s + 6.13e3·27-s + 3.96e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.71·3-s + 0.5·4-s + 0.904·5-s − 1.21·6-s + 1.91·7-s − 0.353·8-s + 1.94·9-s − 0.639·10-s + 0.0820·11-s + 0.857·12-s − 0.823·13-s − 1.35·14-s + 1.55·15-s + 0.250·16-s + 0.552·17-s − 1.37·18-s − 1.21·19-s + 0.452·20-s + 3.28·21-s − 0.0580·22-s + 1.28·23-s − 0.606·24-s − 0.182·25-s + 0.582·26-s + 1.62·27-s + 0.956·28-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 11
Analytic conductor: 86.286486.2864
Root analytic conductor: 9.289059.28905
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 538, ( :5/2), 1)(2,\ 538,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 4.7838856134.783885613
L(12)L(\frac12) ≈\approx 4.7838856134.783885613
L(72)L(\frac{7}{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+4T 1 + 4T
269 1−7.23e4T 1 - 7.23e4T
good3 1−26.7T+243T2 1 - 26.7T + 243T^{2}
5 1−50.5T+3.12e3T2 1 - 50.5T + 3.12e3T^{2}
7 1−248.T+1.68e4T2 1 - 248.T + 1.68e4T^{2}
11 1−32.9T+1.61e5T2 1 - 32.9T + 1.61e5T^{2}
13 1+501.T+3.71e5T2 1 + 501.T + 3.71e5T^{2}
17 1−658.T+1.41e6T2 1 - 658.T + 1.41e6T^{2}
19 1+1.91e3T+2.47e6T2 1 + 1.91e3T + 2.47e6T^{2}
23 1−3.25e3T+6.43e6T2 1 - 3.25e3T + 6.43e6T^{2}
29 1+1.07e3T+2.05e7T2 1 + 1.07e3T + 2.05e7T^{2}
31 1−2.74e3T+2.86e7T2 1 - 2.74e3T + 2.86e7T^{2}
37 1−1.57e4T+6.93e7T2 1 - 1.57e4T + 6.93e7T^{2}
41 1+1.82e4T+1.15e8T2 1 + 1.82e4T + 1.15e8T^{2}
43 1−804.T+1.47e8T2 1 - 804.T + 1.47e8T^{2}
47 1+2.95e3T+2.29e8T2 1 + 2.95e3T + 2.29e8T^{2}
53 1−813.T+4.18e8T2 1 - 813.T + 4.18e8T^{2}
59 1−1.16e4T+7.14e8T2 1 - 1.16e4T + 7.14e8T^{2}
61 1+2.22e4T+8.44e8T2 1 + 2.22e4T + 8.44e8T^{2}
67 1−8.72e3T+1.35e9T2 1 - 8.72e3T + 1.35e9T^{2}
71 1−4.29e4T+1.80e9T2 1 - 4.29e4T + 1.80e9T^{2}
73 1−2.36e4T+2.07e9T2 1 - 2.36e4T + 2.07e9T^{2}
79 1−4.76e4T+3.07e9T2 1 - 4.76e4T + 3.07e9T^{2}
83 1+1.13e5T+3.93e9T2 1 + 1.13e5T + 3.93e9T^{2}
89 1+4.28e3T+5.58e9T2 1 + 4.28e3T + 5.58e9T^{2}
97 1+1.42e5T+8.58e9T2 1 + 1.42e5T + 8.58e9T^{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

−9.765898587512248261852968454545, −9.066293175822729396164605924426, −8.235867854888496458651746233009, −7.83122685348487346051104550558, −6.81073433935763179591345874819, −5.27931742291873261486697931665, −4.25073725091371872303906100888, −2.71946414533525483721781732457, −2.01669771161414398101380547359, −1.26963555233854712671910749804, 1.26963555233854712671910749804, 2.01669771161414398101380547359, 2.71946414533525483721781732457, 4.25073725091371872303906100888, 5.27931742291873261486697931665, 6.81073433935763179591345874819, 7.83122685348487346051104550558, 8.235867854888496458651746233009, 9.066293175822729396164605924426, 9.765898587512248261852968454545

Graph of the ZZ-function along the critical line