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Properties

Label 2-363-1.1-c1-0-16
Degree 22
Conductor 363363
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23·2-s + 3-s + 3.00·4-s + 2·5-s + 2.23·6-s − 4.47·7-s + 2.23·8-s + 9-s + 4.47·10-s + 3.00·12-s − 10.0·14-s + 2·15-s − 0.999·16-s + 4.47·17-s + 2.23·18-s − 4.47·19-s + 6.00·20-s − 4.47·21-s − 4·23-s + 2.23·24-s − 25-s + 27-s − 13.4·28-s + 4.47·29-s + 4.47·30-s − 6.70·32-s + 10.0·34-s + ⋯
L(s)  = 1  + 1.58·2-s + 0.577·3-s + 1.50·4-s + 0.894·5-s + 0.912·6-s − 1.69·7-s + 0.790·8-s + 0.333·9-s + 1.41·10-s + 0.866·12-s − 2.67·14-s + 0.516·15-s − 0.249·16-s + 1.08·17-s + 0.527·18-s − 1.02·19-s + 1.34·20-s − 0.975·21-s − 0.834·23-s + 0.456·24-s − 0.200·25-s + 0.192·27-s − 2.53·28-s + 0.830·29-s + 0.816·30-s − 1.18·32-s + 1.71·34-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: 11
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: 00
Selberg data: (2, 363, ( :1/2), 1)(2,\ 363,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 3.4561946443.456194644
L(12)L(\frac12) ≈\approx 3.4561946443.456194644
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 1−2.23T+2T2 1 - 2.23T + 2T^{2}
5 1−2T+5T2 1 - 2T + 5T^{2}
7 1+4.47T+7T2 1 + 4.47T + 7T^{2}
13 1+13T2 1 + 13T^{2}
17 1−4.47T+17T2 1 - 4.47T + 17T^{2}
19 1+4.47T+19T2 1 + 4.47T + 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−4.47T+29T2 1 - 4.47T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+4.47T+41T2 1 + 4.47T + 41T^{2}
43 1−4.47T+43T2 1 - 4.47T + 43T^{2}
47 1−8T+47T2 1 - 8T + 47T^{2}
53 1−6T+53T2 1 - 6T + 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1−8.94T+61T2 1 - 8.94T + 61T^{2}
67 1+12T+67T2 1 + 12T + 67T^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+8.94T+73T2 1 + 8.94T + 73T^{2}
79 1−13.4T+79T2 1 - 13.4T + 79T^{2}
83 1+8.94T+83T2 1 + 8.94T + 83T^{2}
89 1+14T+89T2 1 + 14T + 89T^{2}
97 1−2T+97T2 1 - 2T + 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

−11.96813214338545798912435604191, −10.38104632610696553786721885962, −9.800274452758984125650845855443, −8.768845110335529202437101245554, −7.21034934891176342175331080954, −6.21714992398416050933747042136, −5.73387068866665816679127942391, −4.23324492548299928706192072988, −3.27526656858308043035087270177, −2.34514472825920689329196705733, 2.34514472825920689329196705733, 3.27526656858308043035087270177, 4.23324492548299928706192072988, 5.73387068866665816679127942391, 6.21714992398416050933747042136, 7.21034934891176342175331080954, 8.768845110335529202437101245554, 9.800274452758984125650845855443, 10.38104632610696553786721885962, 11.96813214338545798912435604191

Graph of the ZZ-function along the critical line