↑

Properties

Label 2-105-1.1-c9-0-6
Degree 22
Conductor 105105
Sign 11
Analytic cond. 54.078754.0787
Root an. cond. 7.353827.35382
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 19.1·2-s − 81·3-s − 146.·4-s + 625·5-s + 1.54e3·6-s − 2.40e3·7-s + 1.25e4·8-s + 6.56e3·9-s − 1.19e4·10-s + 6.69e4·11-s + 1.18e4·12-s − 1.13e5·13-s + 4.58e4·14-s − 5.06e4·15-s − 1.65e5·16-s + 5.03e5·17-s − 1.25e5·18-s − 6.66e5·19-s − 9.17e4·20-s + 1.94e5·21-s − 1.27e6·22-s − 2.33e5·23-s − 1.01e6·24-s + 3.90e5·25-s + 2.16e6·26-s − 5.31e5·27-s + 3.52e5·28-s + ⋯
L(s)  = 1  − 0.844·2-s − 0.577·3-s − 0.286·4-s + 0.447·5-s + 0.487·6-s − 0.377·7-s + 1.08·8-s + 0.333·9-s − 0.377·10-s + 1.37·11-s + 0.165·12-s − 1.09·13-s + 0.319·14-s − 0.258·15-s − 0.631·16-s + 1.46·17-s − 0.281·18-s − 1.17·19-s − 0.128·20-s + 0.218·21-s − 1.16·22-s − 0.173·23-s − 0.627·24-s + 0.200·25-s + 0.928·26-s − 0.192·27-s + 0.108·28-s + ⋯

Functional equation

Λ(s)=(105s/2 ΓC(s) L(s)=( Λ(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}
Λ(s)=(105s/2 ΓC(s+9/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 105105    =    3⋅5⋅73 \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 54.078754.0787
Root analytic conductor: 7.353827.35382
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 105, ( :9/2), 1)(2,\ 105,\ (\ :9/2),\ 1)

Particular Values

L(5)L(5) ≈\approx 0.83503999900.8350399990
L(12)L(\frac12) ≈\approx 0.83503999900.8350399990
L(112)L(\frac{11}{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)
bad3 1+81T 1 + 81T
5 1−625T 1 - 625T
7 1+2.40e3T 1 + 2.40e3T
good2 1+19.1T+512T2 1 + 19.1T + 512T^{2}
11 1−6.69e4T+2.35e9T2 1 - 6.69e4T + 2.35e9T^{2}
13 1+1.13e5T+1.06e10T2 1 + 1.13e5T + 1.06e10T^{2}
17 1−5.03e5T+1.18e11T2 1 - 5.03e5T + 1.18e11T^{2}
19 1+6.66e5T+3.22e11T2 1 + 6.66e5T + 3.22e11T^{2}
23 1+2.33e5T+1.80e12T2 1 + 2.33e5T + 1.80e12T^{2}
29 1+1.70e6T+1.45e13T2 1 + 1.70e6T + 1.45e13T^{2}
31 1+4.60e6T+2.64e13T2 1 + 4.60e6T + 2.64e13T^{2}
37 1−3.33e6T+1.29e14T2 1 - 3.33e6T + 1.29e14T^{2}
41 1+1.57e7T+3.27e14T2 1 + 1.57e7T + 3.27e14T^{2}
43 1+1.97e7T+5.02e14T2 1 + 1.97e7T + 5.02e14T^{2}
47 1−6.65e7T+1.11e15T2 1 - 6.65e7T + 1.11e15T^{2}
53 1+6.63e7T+3.29e15T2 1 + 6.63e7T + 3.29e15T^{2}
59 1−1.18e8T+8.66e15T2 1 - 1.18e8T + 8.66e15T^{2}
61 1+3.11e7T+1.16e16T2 1 + 3.11e7T + 1.16e16T^{2}
67 1+1.76e8T+2.72e16T2 1 + 1.76e8T + 2.72e16T^{2}
71 1−2.87e8T+4.58e16T2 1 - 2.87e8T + 4.58e16T^{2}
73 1−1.47e8T+5.88e16T2 1 - 1.47e8T + 5.88e16T^{2}
79 1−1.80e8T+1.19e17T2 1 - 1.80e8T + 1.19e17T^{2}
83 1−6.04e8T+1.86e17T2 1 - 6.04e8T + 1.86e17T^{2}
89 1−6.67e8T+3.50e17T2 1 - 6.67e8T + 3.50e17T^{2}
97 1−1.69e9T+7.60e17T2 1 - 1.69e9T + 7.60e17T^{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

−11.94221105956604710662524817668, −10.57211575687839843380207332284, −9.759184615089516068602960660019, −9.016011265072199760048413971389, −7.62376916492646723854130347302, −6.49706071587561980565655244627, −5.18955598356575558644080282844, −3.86455223318867957534343988710, −1.81034735931760690852168849796, −0.60729270156525285857178521744, 0.60729270156525285857178521744, 1.81034735931760690852168849796, 3.86455223318867957534343988710, 5.18955598356575558644080282844, 6.49706071587561980565655244627, 7.62376916492646723854130347302, 9.016011265072199760048413971389, 9.759184615089516068602960660019, 10.57211575687839843380207332284, 11.94221105956604710662524817668

Graph of the ZZ-function along the critical line