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Properties

Label 2-538-1.1-c5-0-26
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 − 8.12·3-s + 16·4-s + 20.9·5-s − 32.4·6-s − 59.5·7-s + 64·8-s − 177.·9-s + 83.8·10-s + 652.·11-s − 129.·12-s − 914.·13-s − 238.·14-s − 170.·15-s + 256·16-s − 207.·17-s − 708.·18-s − 535.·19-s + 335.·20-s + 483.·21-s + 2.61e3·22-s + 3.52e3·23-s − 519.·24-s − 2.68e3·25-s − 3.65e3·26-s + 3.41e3·27-s − 952.·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.520·3-s + 0.5·4-s + 0.374·5-s − 0.368·6-s − 0.459·7-s + 0.353·8-s − 0.728·9-s + 0.265·10-s + 1.62·11-s − 0.260·12-s − 1.50·13-s − 0.324·14-s − 0.195·15-s + 0.250·16-s − 0.174·17-s − 0.515·18-s − 0.340·19-s + 0.187·20-s + 0.239·21-s + 1.15·22-s + 1.39·23-s − 0.184·24-s − 0.859·25-s − 1.06·26-s + 0.900·27-s − 0.229·28-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 2.6817438672.681743867
L(12)L(\frac12) ≈\approx 2.6817438672.681743867
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+8.12T+243T2 1 + 8.12T + 243T^{2}
5 1−20.9T+3.12e3T2 1 - 20.9T + 3.12e3T^{2}
7 1+59.5T+1.68e4T2 1 + 59.5T + 1.68e4T^{2}
11 1−652.T+1.61e5T2 1 - 652.T + 1.61e5T^{2}
13 1+914.T+3.71e5T2 1 + 914.T + 3.71e5T^{2}
17 1+207.T+1.41e6T2 1 + 207.T + 1.41e6T^{2}
19 1+535.T+2.47e6T2 1 + 535.T + 2.47e6T^{2}
23 1−3.52e3T+6.43e6T2 1 - 3.52e3T + 6.43e6T^{2}
29 1−24.8T+2.05e7T2 1 - 24.8T + 2.05e7T^{2}
31 1−9.91e3T+2.86e7T2 1 - 9.91e3T + 2.86e7T^{2}
37 1−5.66e3T+6.93e7T2 1 - 5.66e3T + 6.93e7T^{2}
41 1+2.96e3T+1.15e8T2 1 + 2.96e3T + 1.15e8T^{2}
43 1+1.64e4T+1.47e8T2 1 + 1.64e4T + 1.47e8T^{2}
47 1−7.66e3T+2.29e8T2 1 - 7.66e3T + 2.29e8T^{2}
53 1+1.63e4T+4.18e8T2 1 + 1.63e4T + 4.18e8T^{2}
59 1−2.82e4T+7.14e8T2 1 - 2.82e4T + 7.14e8T^{2}
61 1−2.48e4T+8.44e8T2 1 - 2.48e4T + 8.44e8T^{2}
67 1−3.62e4T+1.35e9T2 1 - 3.62e4T + 1.35e9T^{2}
71 1−5.61e4T+1.80e9T2 1 - 5.61e4T + 1.80e9T^{2}
73 1−7.57e4T+2.07e9T2 1 - 7.57e4T + 2.07e9T^{2}
79 1+3.41e4T+3.07e9T2 1 + 3.41e4T + 3.07e9T^{2}
83 1−3.25e4T+3.93e9T2 1 - 3.25e4T + 3.93e9T^{2}
89 1+2.59e4T+5.58e9T2 1 + 2.59e4T + 5.58e9T^{2}
97 1−1.48e5T+8.58e9T2 1 - 1.48e5T + 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

−10.00344709845432443027923473254, −9.374723244681234765771124099752, −8.219609508969567811431764434637, −6.77156821555178630858224275211, −6.47120282126823491157982802144, −5.34624962951574395749898329460, −4.52324614216963712690827272095, −3.28603183019328689792258938989, −2.22230683705874773779849238077, −0.74053517594024765282978609568, 0.74053517594024765282978609568, 2.22230683705874773779849238077, 3.28603183019328689792258938989, 4.52324614216963712690827272095, 5.34624962951574395749898329460, 6.47120282126823491157982802144, 6.77156821555178630858224275211, 8.219609508969567811431764434637, 9.374723244681234765771124099752, 10.00344709845432443027923473254

Graph of the ZZ-function along the critical line