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Properties

Label 2-1338-1.1-c1-0-36
Degree 22
Conductor 13381338
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 0.753·5-s + 6-s − 4.69·7-s + 8-s + 9-s − 0.753·10-s − 5.51·11-s + 12-s − 4.13·13-s − 4.69·14-s − 0.753·15-s + 16-s + 4.34·17-s + 18-s + 5.65·19-s − 0.753·20-s − 4.69·21-s − 5.51·22-s − 5.69·23-s + 24-s − 4.43·25-s − 4.13·26-s + 27-s − 4.69·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.336·5-s + 0.408·6-s − 1.77·7-s + 0.353·8-s + 0.333·9-s − 0.238·10-s − 1.66·11-s + 0.288·12-s − 1.14·13-s − 1.25·14-s − 0.194·15-s + 0.250·16-s + 1.05·17-s + 0.235·18-s + 1.29·19-s − 0.168·20-s − 1.02·21-s − 1.17·22-s − 1.18·23-s + 0.204·24-s − 0.886·25-s − 0.811·26-s + 0.192·27-s − 0.886·28-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: −1-1
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: 11
Selberg data: (2, 1338, ( :1/2), −1)(2,\ 1338,\ (\ :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
223 1+T 1 + T
good5 1+0.753T+5T2 1 + 0.753T + 5T^{2}
7 1+4.69T+7T2 1 + 4.69T + 7T^{2}
11 1+5.51T+11T2 1 + 5.51T + 11T^{2}
13 1+4.13T+13T2 1 + 4.13T + 13T^{2}
17 1−4.34T+17T2 1 - 4.34T + 17T^{2}
19 1−5.65T+19T2 1 - 5.65T + 19T^{2}
23 1+5.69T+23T2 1 + 5.69T + 23T^{2}
29 1+0.417T+29T2 1 + 0.417T + 29T^{2}
31 1+1.55T+31T2 1 + 1.55T + 31T^{2}
37 1+8.89T+37T2 1 + 8.89T + 37T^{2}
41 1+9.36T+41T2 1 + 9.36T + 41T^{2}
43 1−1.04T+43T2 1 - 1.04T + 43T^{2}
47 1−2.57T+47T2 1 - 2.57T + 47T^{2}
53 1+8.32T+53T2 1 + 8.32T + 53T^{2}
59 1−13.0T+59T2 1 - 13.0T + 59T^{2}
61 1+15.5T+61T2 1 + 15.5T + 61T^{2}
67 1−11.5T+67T2 1 - 11.5T + 67T^{2}
71 1−10.0T+71T2 1 - 10.0T + 71T^{2}
73 1−1.53T+73T2 1 - 1.53T + 73T^{2}
79 1−6.46T+79T2 1 - 6.46T + 79T^{2}
83 1−2.93T+83T2 1 - 2.93T + 83T^{2}
89 1+4.59T+89T2 1 + 4.59T + 89T^{2}
97 1−3.37T+97T2 1 - 3.37T + 97T^{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

−9.567304693105465161281847061193, −8.114976407795153467770668087375, −7.54750257708547906537849086359, −6.83632483959898722439332566181, −5.69450475349190432866064007291, −5.08256965940310810458814162038, −3.66274583474964439446445294686, −3.18437186873917280734313608602, −2.27842930446747794927817682402, 0, 2.27842930446747794927817682402, 3.18437186873917280734313608602, 3.66274583474964439446445294686, 5.08256965940310810458814162038, 5.69450475349190432866064007291, 6.83632483959898722439332566181, 7.54750257708547906537849086359, 8.114976407795153467770668087375, 9.567304693105465161281847061193

Graph of the ZZ-function along the critical line