↑

Properties

Label 2-4002-1.1-c1-0-46
Degree 22
Conductor 40024002
Sign −1-1
Analytic cond. 31.956131.9561
Root an. cond. 5.652975.65297
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 0.662·5-s + 6-s − 0.537·7-s − 8-s + 9-s + 0.662·10-s + 1.91·11-s − 12-s − 2.29·13-s + 0.537·14-s + 0.662·15-s + 16-s − 1.24·17-s − 18-s + 2.15·19-s − 0.662·20-s + 0.537·21-s − 1.91·22-s + 23-s + 24-s − 4.56·25-s + 2.29·26-s − 27-s − 0.537·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.296·5-s + 0.408·6-s − 0.203·7-s − 0.353·8-s + 0.333·9-s + 0.209·10-s + 0.576·11-s − 0.288·12-s − 0.635·13-s + 0.143·14-s + 0.171·15-s + 0.250·16-s − 0.303·17-s − 0.235·18-s + 0.493·19-s − 0.148·20-s + 0.117·21-s − 0.407·22-s + 0.208·23-s + 0.204·24-s − 0.912·25-s + 0.449·26-s − 0.192·27-s − 0.101·28-s + ⋯

Functional equation

Λ(s)=(4002s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4002s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 40024002    =    2⋅3⋅23⋅292 \cdot 3 \cdot 23 \cdot 29
Sign: −1-1
Analytic conductor: 31.956131.9561
Root analytic conductor: 5.652975.65297
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4002, ( :1/2), −1)(2,\ 4002,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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+T 1 + T
3 1+T 1 + T
23 1−T 1 - T
29 1−T 1 - T
good5 1+0.662T+5T2 1 + 0.662T + 5T^{2}
7 1+0.537T+7T2 1 + 0.537T + 7T^{2}
11 1−1.91T+11T2 1 - 1.91T + 11T^{2}
13 1+2.29T+13T2 1 + 2.29T + 13T^{2}
17 1+1.24T+17T2 1 + 1.24T + 17T^{2}
19 1−2.15T+19T2 1 - 2.15T + 19T^{2}
31 1−4.51T+31T2 1 - 4.51T + 31T^{2}
37 1+4.68T+37T2 1 + 4.68T + 37T^{2}
41 1−3.06T+41T2 1 - 3.06T + 41T^{2}
43 1−3.78T+43T2 1 - 3.78T + 43T^{2}
47 1+3.25T+47T2 1 + 3.25T + 47T^{2}
53 1+1.37T+53T2 1 + 1.37T + 53T^{2}
59 1−10.9T+59T2 1 - 10.9T + 59T^{2}
61 1−0.986T+61T2 1 - 0.986T + 61T^{2}
67 1+13.5T+67T2 1 + 13.5T + 67T^{2}
71 1−4.39T+71T2 1 - 4.39T + 71T^{2}
73 1−6.04T+73T2 1 - 6.04T + 73T^{2}
79 1−0.602T+79T2 1 - 0.602T + 79T^{2}
83 1+5.40T+83T2 1 + 5.40T + 83T^{2}
89 1+1.27T+89T2 1 + 1.27T + 89T^{2}
97 1−10.7T+97T2 1 - 10.7T + 97T^{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

−8.049409319528013506824475400728, −7.35600982146380516398334985665, −6.71282273228537161894178353572, −6.03977870881608757263508343353, −5.15764514691083298485065694326, −4.29636515836982978500304164785, −3.36680924151244308604155666244, −2.29687088851393736990536033869, −1.17405488524144781838885095248, 0, 1.17405488524144781838885095248, 2.29687088851393736990536033869, 3.36680924151244308604155666244, 4.29636515836982978500304164785, 5.15764514691083298485065694326, 6.03977870881608757263508343353, 6.71282273228537161894178353572, 7.35600982146380516398334985665, 8.049409319528013506824475400728

Graph of the ZZ-function along the critical line