↑

Properties

Label 2-538-1.1-c5-0-87
Degree 22
Conductor 538538
Sign −1-1
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 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 9.07·3-s + 16·4-s + 62.0·5-s − 36.2·6-s − 113.·7-s − 64·8-s − 160.·9-s − 248.·10-s + 263.·11-s + 145.·12-s − 273.·13-s + 453.·14-s + 562.·15-s + 256·16-s − 857.·17-s + 642.·18-s + 1.13e3·19-s + 992.·20-s − 1.02e3·21-s − 1.05e3·22-s + 2.42e3·23-s − 580.·24-s + 721.·25-s + 1.09e3·26-s − 3.66e3·27-s − 1.81e3·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.581·3-s + 0.5·4-s + 1.10·5-s − 0.411·6-s − 0.874·7-s − 0.353·8-s − 0.661·9-s − 0.784·10-s + 0.656·11-s + 0.290·12-s − 0.448·13-s + 0.618·14-s + 0.645·15-s + 0.250·16-s − 0.719·17-s + 0.467·18-s + 0.720·19-s + 0.554·20-s − 0.509·21-s − 0.463·22-s + 0.956·23-s − 0.205·24-s + 0.230·25-s + 0.317·26-s − 0.966·27-s − 0.437·28-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: −1-1
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: 11
Selberg data: (2, 538, ( :5/2), −1)(2,\ 538,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
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 1−9.07T+243T2 1 - 9.07T + 243T^{2}
5 1−62.0T+3.12e3T2 1 - 62.0T + 3.12e3T^{2}
7 1+113.T+1.68e4T2 1 + 113.T + 1.68e4T^{2}
11 1−263.T+1.61e5T2 1 - 263.T + 1.61e5T^{2}
13 1+273.T+3.71e5T2 1 + 273.T + 3.71e5T^{2}
17 1+857.T+1.41e6T2 1 + 857.T + 1.41e6T^{2}
19 1−1.13e3T+2.47e6T2 1 - 1.13e3T + 2.47e6T^{2}
23 1−2.42e3T+6.43e6T2 1 - 2.42e3T + 6.43e6T^{2}
29 1−7.69e3T+2.05e7T2 1 - 7.69e3T + 2.05e7T^{2}
31 1+4.92e3T+2.86e7T2 1 + 4.92e3T + 2.86e7T^{2}
37 1−757.T+6.93e7T2 1 - 757.T + 6.93e7T^{2}
41 1+199.T+1.15e8T2 1 + 199.T + 1.15e8T^{2}
43 1+1.12e4T+1.47e8T2 1 + 1.12e4T + 1.47e8T^{2}
47 1+1.82e4T+2.29e8T2 1 + 1.82e4T + 2.29e8T^{2}
53 1+1.96e4T+4.18e8T2 1 + 1.96e4T + 4.18e8T^{2}
59 1−1.45e4T+7.14e8T2 1 - 1.45e4T + 7.14e8T^{2}
61 1+5.38e4T+8.44e8T2 1 + 5.38e4T + 8.44e8T^{2}
67 1+3.72e4T+1.35e9T2 1 + 3.72e4T + 1.35e9T^{2}
71 1−4.75e4T+1.80e9T2 1 - 4.75e4T + 1.80e9T^{2}
73 1−8.82e3T+2.07e9T2 1 - 8.82e3T + 2.07e9T^{2}
79 1−2.95e4T+3.07e9T2 1 - 2.95e4T + 3.07e9T^{2}
83 1+559.T+3.93e9T2 1 + 559.T + 3.93e9T^{2}
89 1−1.00e5T+5.58e9T2 1 - 1.00e5T + 5.58e9T^{2}
97 1+1.19e5T+8.58e9T2 1 + 1.19e5T + 8.58e9T^{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

−9.385443298159200445221790767497, −9.053002483220135473973233796849, −7.993385624632637651359826785639, −6.77780598692035858434833035937, −6.21809440976821910443587537024, −5.05268355107760886811231791610, −3.33770336010769079915134167175, −2.57176511059207811057052481746, −1.45450421511945341471440460090, 0, 1.45450421511945341471440460090, 2.57176511059207811057052481746, 3.33770336010769079915134167175, 5.05268355107760886811231791610, 6.21809440976821910443587537024, 6.77780598692035858434833035937, 7.993385624632637651359826785639, 9.053002483220135473973233796849, 9.385443298159200445221790767497

Graph of the ZZ-function along the critical line