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Properties

Label 2-538-1.1-c3-0-16
Degree 22
Conductor 538538
Sign 11
Analytic cond. 31.743031.7430
Root an. cond. 5.634095.63409
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 8.60·3-s + 4·4-s + 0.977·5-s + 17.2·6-s + 26.8·7-s − 8·8-s + 47.0·9-s − 1.95·10-s + 37.8·11-s − 34.4·12-s + 67.8·13-s − 53.6·14-s − 8.41·15-s + 16·16-s + 26.7·17-s − 94.1·18-s + 52.8·19-s + 3.91·20-s − 230.·21-s − 75.6·22-s + 103.·23-s + 68.8·24-s − 124.·25-s − 135.·26-s − 172.·27-s + 107.·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.65·3-s + 0.5·4-s + 0.0874·5-s + 1.17·6-s + 1.44·7-s − 0.353·8-s + 1.74·9-s − 0.0618·10-s + 1.03·11-s − 0.828·12-s + 1.44·13-s − 1.02·14-s − 0.144·15-s + 0.250·16-s + 0.381·17-s − 1.23·18-s + 0.638·19-s + 0.0437·20-s − 2.39·21-s − 0.732·22-s + 0.939·23-s + 0.585·24-s − 0.992·25-s − 1.02·26-s − 1.23·27-s + 0.724·28-s + ⋯

Functional equation

Λ(s)=(538s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(538s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 538538    =    2⋅2692 \cdot 269
Sign: 11
Analytic conductor: 31.743031.7430
Root analytic conductor: 5.634095.63409
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 538, ( :3/2), 1)(2,\ 538,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.1860214911.186021491
L(12)L(\frac12) ≈\approx 1.1860214911.186021491
L(52)L(\frac{5}{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+2T 1 + 2T
269 1+269T 1 + 269T
good3 1+8.60T+27T2 1 + 8.60T + 27T^{2}
5 1−0.977T+125T2 1 - 0.977T + 125T^{2}
7 1−26.8T+343T2 1 - 26.8T + 343T^{2}
11 1−37.8T+1.33e3T2 1 - 37.8T + 1.33e3T^{2}
13 1−67.8T+2.19e3T2 1 - 67.8T + 2.19e3T^{2}
17 1−26.7T+4.91e3T2 1 - 26.7T + 4.91e3T^{2}
19 1−52.8T+6.85e3T2 1 - 52.8T + 6.85e3T^{2}
23 1−103.T+1.21e4T2 1 - 103.T + 1.21e4T^{2}
29 1−44.7T+2.43e4T2 1 - 44.7T + 2.43e4T^{2}
31 1+126.T+2.97e4T2 1 + 126.T + 2.97e4T^{2}
37 1+63.7T+5.06e4T2 1 + 63.7T + 5.06e4T^{2}
41 1−474.T+6.89e4T2 1 - 474.T + 6.89e4T^{2}
43 1+365.T+7.95e4T2 1 + 365.T + 7.95e4T^{2}
47 1+146.T+1.03e5T2 1 + 146.T + 1.03e5T^{2}
53 1−201.T+1.48e5T2 1 - 201.T + 1.48e5T^{2}
59 1+36.7T+2.05e5T2 1 + 36.7T + 2.05e5T^{2}
61 1+231.T+2.26e5T2 1 + 231.T + 2.26e5T^{2}
67 1−561.T+3.00e5T2 1 - 561.T + 3.00e5T^{2}
71 1−866.T+3.57e5T2 1 - 866.T + 3.57e5T^{2}
73 1−19.0T+3.89e5T2 1 - 19.0T + 3.89e5T^{2}
79 1+925.T+4.93e5T2 1 + 925.T + 4.93e5T^{2}
83 1−1.28e3T+5.71e5T2 1 - 1.28e3T + 5.71e5T^{2}
89 1+245.T+7.04e5T2 1 + 245.T + 7.04e5T^{2}
97 1+1.21e3T+9.12e5T2 1 + 1.21e3T + 9.12e5T^{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

−10.74154442230112592995237556322, −9.656429162741775003919539801626, −8.657844692267166008093178427562, −7.67572615483943247196251646406, −6.68683689596583351418894437764, −5.85216271616979485384299874929, −5.05006444359746864079907484517, −3.85641882587468177664289946305, −1.57380169961291688528678283148, −0.920809298679629953240507495640, 0.920809298679629953240507495640, 1.57380169961291688528678283148, 3.85641882587468177664289946305, 5.05006444359746864079907484517, 5.85216271616979485384299874929, 6.68683689596583351418894437764, 7.67572615483943247196251646406, 8.657844692267166008093178427562, 9.656429162741775003919539801626, 10.74154442230112592995237556322

Graph of the ZZ-function along the critical line