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Properties

Label 2-105-1.1-c3-0-8
Degree 22
Conductor 105105
Sign 11
Analytic cond. 6.195206.19520
Root an. cond. 2.489012.48901
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  + 4.70·2-s + 3·3-s + 14.1·4-s − 5·5-s + 14.1·6-s + 7·7-s + 28.7·8-s + 9·9-s − 23.5·10-s + 24.5·11-s + 42.3·12-s − 35.0·13-s + 32.9·14-s − 15·15-s + 22.1·16-s − 18.4·17-s + 42.3·18-s − 67.4·19-s − 70.5·20-s + 21·21-s + 115.·22-s − 145.·23-s + 86.1·24-s + 25·25-s − 164.·26-s + 27·27-s + 98.7·28-s + ⋯
L(s)  = 1  + 1.66·2-s + 0.577·3-s + 1.76·4-s − 0.447·5-s + 0.959·6-s + 0.377·7-s + 1.26·8-s + 0.333·9-s − 0.743·10-s + 0.674·11-s + 1.01·12-s − 0.747·13-s + 0.628·14-s − 0.258·15-s + 0.345·16-s − 0.262·17-s + 0.554·18-s − 0.813·19-s − 0.788·20-s + 0.218·21-s + 1.12·22-s − 1.32·23-s + 0.732·24-s + 0.200·25-s − 1.24·26-s + 0.192·27-s + 0.666·28-s + ⋯

Functional equation

Λ(s)=(105s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(105s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, 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: 6.195206.19520
Root analytic conductor: 2.489012.48901
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 105, ( :3/2), 1)(2,\ 105,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 4.1887686994.188768699
L(12)L(\frac12) ≈\approx 4.1887686994.188768699
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)
bad3 1−3T 1 - 3T
5 1+5T 1 + 5T
7 1−7T 1 - 7T
good2 1−4.70T+8T2 1 - 4.70T + 8T^{2}
11 1−24.5T+1.33e3T2 1 - 24.5T + 1.33e3T^{2}
13 1+35.0T+2.19e3T2 1 + 35.0T + 2.19e3T^{2}
17 1+18.4T+4.91e3T2 1 + 18.4T + 4.91e3T^{2}
19 1+67.4T+6.85e3T2 1 + 67.4T + 6.85e3T^{2}
23 1+145.T+1.21e4T2 1 + 145.T + 1.21e4T^{2}
29 1−214.T+2.43e4T2 1 - 214.T + 2.43e4T^{2}
31 1+88.6T+2.97e4T2 1 + 88.6T + 2.97e4T^{2}
37 1−162.T+5.06e4T2 1 - 162.T + 5.06e4T^{2}
41 1+337.T+6.89e4T2 1 + 337.T + 6.89e4T^{2}
43 1−122.T+7.95e4T2 1 - 122.T + 7.95e4T^{2}
47 1−354.T+1.03e5T2 1 - 354.T + 1.03e5T^{2}
53 1−676.T+1.48e5T2 1 - 676.T + 1.48e5T^{2}
59 1−501.T+2.05e5T2 1 - 501.T + 2.05e5T^{2}
61 1+708.T+2.26e5T2 1 + 708.T + 2.26e5T^{2}
67 1+907.T+3.00e5T2 1 + 907.T + 3.00e5T^{2}
71 1−430.T+3.57e5T2 1 - 430.T + 3.57e5T^{2}
73 1−41.3T+3.89e5T2 1 - 41.3T + 3.89e5T^{2}
79 1−890.T+4.93e5T2 1 - 890.T + 4.93e5T^{2}
83 1+1.05e3T+5.71e5T2 1 + 1.05e3T + 5.71e5T^{2}
89 1−1.47e3T+7.04e5T2 1 - 1.47e3T + 7.04e5T^{2}
97 1−555.T+9.12e5T2 1 - 555.T + 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

−13.45661982419435445974030247562, −12.31294012545683469886175642022, −11.74340904356697436312554036667, −10.38833133752295853137547898109, −8.762382598156180586820762710996, −7.41717170158120664380132296828, −6.24001980998692394717884292756, −4.71333953154907923884078271650, −3.82511888816381233528132119600, −2.32086255885988398607874075783, 2.32086255885988398607874075783, 3.82511888816381233528132119600, 4.71333953154907923884078271650, 6.24001980998692394717884292756, 7.41717170158120664380132296828, 8.762382598156180586820762710996, 10.38833133752295853137547898109, 11.74340904356697436312554036667, 12.31294012545683469886175642022, 13.45661982419435445974030247562

Graph of the ZZ-function along the critical line