↑

Properties

Label 2-4002-1.1-c1-0-44
Degree 22
Conductor 40024002
Sign 11
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 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 3.49·5-s − 6-s − 1.05·7-s + 8-s + 9-s + 3.49·10-s + 4.23·11-s − 12-s + 5.43·13-s − 1.05·14-s − 3.49·15-s + 16-s − 3.65·17-s + 18-s − 1.05·19-s + 3.49·20-s + 1.05·21-s + 4.23·22-s + 23-s − 24-s + 7.21·25-s + 5.43·26-s − 27-s − 1.05·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.56·5-s − 0.408·6-s − 0.399·7-s + 0.353·8-s + 0.333·9-s + 1.10·10-s + 1.27·11-s − 0.288·12-s + 1.50·13-s − 0.282·14-s − 0.902·15-s + 0.250·16-s − 0.887·17-s + 0.235·18-s − 0.242·19-s + 0.781·20-s + 0.230·21-s + 0.903·22-s + 0.208·23-s − 0.204·24-s + 1.44·25-s + 1.06·26-s − 0.192·27-s − 0.199·28-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: 11
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: 00
Selberg data: (2, 4002, ( :1/2), 1)(2,\ 4002,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.8520213263.852021326
L(12)L(\frac12) ≈\approx 3.8520213263.852021326
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 1−3.49T+5T2 1 - 3.49T + 5T^{2}
7 1+1.05T+7T2 1 + 1.05T + 7T^{2}
11 1−4.23T+11T2 1 - 4.23T + 11T^{2}
13 1−5.43T+13T2 1 - 5.43T + 13T^{2}
17 1+3.65T+17T2 1 + 3.65T + 17T^{2}
19 1+1.05T+19T2 1 + 1.05T + 19T^{2}
31 1+4.03T+31T2 1 + 4.03T + 31T^{2}
37 1−7.11T+37T2 1 - 7.11T + 37T^{2}
41 1−4.28T+41T2 1 - 4.28T + 41T^{2}
43 1−8.48T+43T2 1 - 8.48T + 43T^{2}
47 1+4.96T+47T2 1 + 4.96T + 47T^{2}
53 1−9.06T+53T2 1 - 9.06T + 53T^{2}
59 1+13.2T+59T2 1 + 13.2T + 59T^{2}
61 1+6.93T+61T2 1 + 6.93T + 61T^{2}
67 1+12.9T+67T2 1 + 12.9T + 67T^{2}
71 1−9.18T+71T2 1 - 9.18T + 71T^{2}
73 1+12.3T+73T2 1 + 12.3T + 73T^{2}
79 1−4.66T+79T2 1 - 4.66T + 79T^{2}
83 1+4.60T+83T2 1 + 4.60T + 83T^{2}
89 1+13.8T+89T2 1 + 13.8T + 89T^{2}
97 1−4.83T+97T2 1 - 4.83T + 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

−8.718957115725790251562882155576, −7.38221378881649191888927957758, −6.50279410499102566864086594319, −6.11815993859787658002547032110, −5.81636428238400712831404758654, −4.67415891929410516698011682525, −4.01144486594651197531888680076, −2.99835264138898912000984326483, −1.90225615557712198814826565915, −1.16256757955017448808660945585, 1.16256757955017448808660945585, 1.90225615557712198814826565915, 2.99835264138898912000984326483, 4.01144486594651197531888680076, 4.67415891929410516698011682525, 5.81636428238400712831404758654, 6.11815993859787658002547032110, 6.50279410499102566864086594319, 7.38221378881649191888927957758, 8.718957115725790251562882155576

Graph of the ZZ-function along the critical line