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Properties

Label 2-363-1.1-c1-0-17
Degree 22
Conductor 363363
Sign −1-1
Analytic cond. 2.898562.89856
Root an. cond. 1.702511.70251
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  + 0.618·2-s + 3-s − 1.61·4-s − 2.61·5-s + 0.618·6-s − 3·7-s − 2.23·8-s + 9-s − 1.61·10-s − 1.61·12-s − 1.76·13-s − 1.85·14-s − 2.61·15-s + 1.85·16-s − 1.61·17-s + 0.618·18-s − 5.85·19-s + 4.23·20-s − 3·21-s + 3.47·23-s − 2.23·24-s + 1.85·25-s − 1.09·26-s + 27-s + 4.85·28-s − 4.47·29-s − 1.61·30-s + ⋯
L(s)  = 1  + 0.437·2-s + 0.577·3-s − 0.809·4-s − 1.17·5-s + 0.252·6-s − 1.13·7-s − 0.790·8-s + 0.333·9-s − 0.511·10-s − 0.467·12-s − 0.489·13-s − 0.495·14-s − 0.675·15-s + 0.463·16-s − 0.392·17-s + 0.145·18-s − 1.34·19-s + 0.947·20-s − 0.654·21-s + 0.723·23-s − 0.456·24-s + 0.370·25-s − 0.213·26-s + 0.192·27-s + 0.917·28-s − 0.830·29-s − 0.295·30-s + ⋯

Functional equation

Λ(s)=(363s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(363s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 363363    =    3⋅1123 \cdot 11^{2}
Sign: −1-1
Analytic conductor: 2.898562.89856
Root analytic conductor: 1.702511.70251
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 363, ( :1/2), −1)(2,\ 363,\ (\ :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)
bad3 1−T 1 - T
11 1 1
good2 1−0.618T+2T2 1 - 0.618T + 2T^{2}
5 1+2.61T+5T2 1 + 2.61T + 5T^{2}
7 1+3T+7T2 1 + 3T + 7T^{2}
13 1+1.76T+13T2 1 + 1.76T + 13T^{2}
17 1+1.61T+17T2 1 + 1.61T + 17T^{2}
19 1+5.85T+19T2 1 + 5.85T + 19T^{2}
23 1−3.47T+23T2 1 - 3.47T + 23T^{2}
29 1+4.47T+29T2 1 + 4.47T + 29T^{2}
31 1−2.85T+31T2 1 - 2.85T + 31T^{2}
37 1−0.236T+37T2 1 - 0.236T + 37T^{2}
41 1−11.9T+41T2 1 - 11.9T + 41T^{2}
43 1+6.23T+43T2 1 + 6.23T + 43T^{2}
47 1−1.61T+47T2 1 - 1.61T + 47T^{2}
53 1+9.61T+53T2 1 + 9.61T + 53T^{2}
59 1−10.3T+59T2 1 - 10.3T + 59T^{2}
61 1+7.85T+61T2 1 + 7.85T + 61T^{2}
67 1+9.56T+67T2 1 + 9.56T + 67T^{2}
71 1+5.56T+71T2 1 + 5.56T + 71T^{2}
73 1−3.23T+73T2 1 - 3.23T + 73T^{2}
79 1+9.47T+79T2 1 + 9.47T + 79T^{2}
83 1+0.708T+83T2 1 + 0.708T + 83T^{2}
89 1−0.527T+89T2 1 - 0.527T + 89T^{2}
97 1+14.0T+97T2 1 + 14.0T + 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

−11.03365687565380218311807407594, −9.853554053915169086297688218088, −9.072922961471080520082514363765, −8.262328614028359519534830216125, −7.23902800568546240059971405639, −6.12188962970225857637937319661, −4.61258388867120864853280197895, −3.86764959082198904621871401652, −2.88545461561253685689728106456, 0, 2.88545461561253685689728106456, 3.86764959082198904621871401652, 4.61258388867120864853280197895, 6.12188962970225857637937319661, 7.23902800568546240059971405639, 8.262328614028359519534830216125, 9.072922961471080520082514363765, 9.853554053915169086297688218088, 11.03365687565380218311807407594

Graph of the ZZ-function along the critical line