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Properties

Label 2-538-1.1-c3-0-51
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 + 10.0·3-s + 4·4-s + 8.61·5-s + 20.0·6-s − 13.5·7-s + 8·8-s + 73.0·9-s + 17.2·10-s + 28.7·11-s + 40.0·12-s + 57.0·13-s − 27.1·14-s + 86.1·15-s + 16·16-s − 79.9·17-s + 146.·18-s − 123.·19-s + 34.4·20-s − 135.·21-s + 57.5·22-s − 32.1·23-s + 80.0·24-s − 50.7·25-s + 114.·26-s + 460.·27-s − 54.3·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.92·3-s + 0.5·4-s + 0.770·5-s + 1.36·6-s − 0.733·7-s + 0.353·8-s + 2.70·9-s + 0.544·10-s + 0.789·11-s + 0.962·12-s + 1.21·13-s − 0.518·14-s + 1.48·15-s + 0.250·16-s − 1.14·17-s + 1.91·18-s − 1.49·19-s + 0.385·20-s − 1.41·21-s + 0.558·22-s − 0.291·23-s + 0.680·24-s − 0.406·25-s + 0.860·26-s + 3.27·27-s − 0.366·28-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 6.7554316056.755431605
L(12)L(\frac12) ≈\approx 6.7554316056.755431605
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 1−10.0T+27T2 1 - 10.0T + 27T^{2}
5 1−8.61T+125T2 1 - 8.61T + 125T^{2}
7 1+13.5T+343T2 1 + 13.5T + 343T^{2}
11 1−28.7T+1.33e3T2 1 - 28.7T + 1.33e3T^{2}
13 1−57.0T+2.19e3T2 1 - 57.0T + 2.19e3T^{2}
17 1+79.9T+4.91e3T2 1 + 79.9T + 4.91e3T^{2}
19 1+123.T+6.85e3T2 1 + 123.T + 6.85e3T^{2}
23 1+32.1T+1.21e4T2 1 + 32.1T + 1.21e4T^{2}
29 1+267.T+2.43e4T2 1 + 267.T + 2.43e4T^{2}
31 1+208.T+2.97e4T2 1 + 208.T + 2.97e4T^{2}
37 1−150.T+5.06e4T2 1 - 150.T + 5.06e4T^{2}
41 1−326.T+6.89e4T2 1 - 326.T + 6.89e4T^{2}
43 1−111.T+7.95e4T2 1 - 111.T + 7.95e4T^{2}
47 1−133.T+1.03e5T2 1 - 133.T + 1.03e5T^{2}
53 1−356.T+1.48e5T2 1 - 356.T + 1.48e5T^{2}
59 1+303.T+2.05e5T2 1 + 303.T + 2.05e5T^{2}
61 1−567.T+2.26e5T2 1 - 567.T + 2.26e5T^{2}
67 1−40.4T+3.00e5T2 1 - 40.4T + 3.00e5T^{2}
71 1+222.T+3.57e5T2 1 + 222.T + 3.57e5T^{2}
73 1+816.T+3.89e5T2 1 + 816.T + 3.89e5T^{2}
79 1−419.T+4.93e5T2 1 - 419.T + 4.93e5T^{2}
83 1+238.T+5.71e5T2 1 + 238.T + 5.71e5T^{2}
89 1−1.07e3T+7.04e5T2 1 - 1.07e3T + 7.04e5T^{2}
97 1−1.20e3T+9.12e5T2 1 - 1.20e3T + 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

−10.25349858252450358456894296230, −9.134618760643616249972060024343, −8.996757049523995618693885163087, −7.74689152132039213379466816284, −6.70709839665970894271122033566, −5.99152921776377006773536980692, −4.12012795841667039327487157973, −3.74563845988830867774497259019, −2.44068385508487015405874461248, −1.72318425283666642663390960677, 1.72318425283666642663390960677, 2.44068385508487015405874461248, 3.74563845988830867774497259019, 4.12012795841667039327487157973, 5.99152921776377006773536980692, 6.70709839665970894271122033566, 7.74689152132039213379466816284, 8.996757049523995618693885163087, 9.134618760643616249972060024343, 10.25349858252450358456894296230

Graph of the ZZ-function along the critical line