↑

Properties

Label 2-1338-1.1-c1-0-10
Degree 22
Conductor 13381338
Sign 11
Analytic cond. 10.683910.6839
Root an. cond. 3.268633.26863
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 − 0.603·5-s − 6-s + 4.55·7-s + 8-s + 9-s − 0.603·10-s + 1.88·11-s − 12-s + 0.332·13-s + 4.55·14-s + 0.603·15-s + 16-s − 1.68·17-s + 18-s + 2.93·19-s − 0.603·20-s − 4.55·21-s + 1.88·22-s − 3.43·23-s − 24-s − 4.63·25-s + 0.332·26-s − 27-s + 4.55·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.269·5-s − 0.408·6-s + 1.71·7-s + 0.353·8-s + 0.333·9-s − 0.190·10-s + 0.568·11-s − 0.288·12-s + 0.0923·13-s + 1.21·14-s + 0.155·15-s + 0.250·16-s − 0.408·17-s + 0.235·18-s + 0.672·19-s − 0.134·20-s − 0.993·21-s + 0.401·22-s − 0.715·23-s − 0.204·24-s − 0.927·25-s + 0.0652·26-s − 0.192·27-s + 0.859·28-s + ⋯

Functional equation

Λ(s)=(1338s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1338s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13381338    =    2⋅3⋅2232 \cdot 3 \cdot 223
Sign: 11
Analytic conductor: 10.683910.6839
Root analytic conductor: 3.268633.26863
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1338, ( :1/2), 1)(2,\ 1338,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6015154712.601515471
L(12)L(\frac12) ≈\approx 2.6015154712.601515471
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
223 1+T 1 + T
good5 1+0.603T+5T2 1 + 0.603T + 5T^{2}
7 1−4.55T+7T2 1 - 4.55T + 7T^{2}
11 1−1.88T+11T2 1 - 1.88T + 11T^{2}
13 1−0.332T+13T2 1 - 0.332T + 13T^{2}
17 1+1.68T+17T2 1 + 1.68T + 17T^{2}
19 1−2.93T+19T2 1 - 2.93T + 19T^{2}
23 1+3.43T+23T2 1 + 3.43T + 23T^{2}
29 1+1.00T+29T2 1 + 1.00T + 29T^{2}
31 1−4.33T+31T2 1 - 4.33T + 31T^{2}
37 1+4.29T+37T2 1 + 4.29T + 37T^{2}
41 1−3.48T+41T2 1 - 3.48T + 41T^{2}
43 1−12.6T+43T2 1 - 12.6T + 43T^{2}
47 1−10.1T+47T2 1 - 10.1T + 47T^{2}
53 1−3.19T+53T2 1 - 3.19T + 53T^{2}
59 1+11.9T+59T2 1 + 11.9T + 59T^{2}
61 1−5.00T+61T2 1 - 5.00T + 61T^{2}
67 1+0.731T+67T2 1 + 0.731T + 67T^{2}
71 1−2.34T+71T2 1 - 2.34T + 71T^{2}
73 1−11.5T+73T2 1 - 11.5T + 73T^{2}
79 1+6.48T+79T2 1 + 6.48T + 79T^{2}
83 1+11.6T+83T2 1 + 11.6T + 83T^{2}
89 1+8.85T+89T2 1 + 8.85T + 89T^{2}
97 1−7.27T+97T2 1 - 7.27T + 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

−9.734136585398062783576585854837, −8.664966153588981180912579827929, −7.78915124849568855131686130105, −7.20888155315302884164913742504, −6.05093966177539974703912322078, −5.40070458532665873952805529729, −4.45926339529163598139365247668, −3.94273131590734835766375946953, −2.32398238771599725211503910593, −1.22520287038283028130514084472, 1.22520287038283028130514084472, 2.32398238771599725211503910593, 3.94273131590734835766375946953, 4.45926339529163598139365247668, 5.40070458532665873952805529729, 6.05093966177539974703912322078, 7.20888155315302884164913742504, 7.78915124849568855131686130105, 8.664966153588981180912579827929, 9.734136585398062783576585854837

Graph of the ZZ-function along the critical line