↑

Properties

Label 2-363-1.1-c1-0-9
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.792·2-s + 3-s − 1.37·4-s + 3.37·5-s + 0.792·6-s + 2.52·7-s − 2.67·8-s + 9-s + 2.67·10-s − 1.37·12-s − 5.84·13-s + 2·14-s + 3.37·15-s + 0.627·16-s + 2.67·17-s + 0.792·18-s − 0.939·19-s − 4.62·20-s + 2.52·21-s + 2·23-s − 2.67·24-s + 6.37·25-s − 4.62·26-s + 27-s − 3.46·28-s − 0.792·29-s + 2.67·30-s + ⋯
L(s)  = 1  + 0.560·2-s + 0.577·3-s − 0.686·4-s + 1.50·5-s + 0.323·6-s + 0.954·7-s − 0.944·8-s + 0.333·9-s + 0.844·10-s − 0.396·12-s − 1.61·13-s + 0.534·14-s + 0.870·15-s + 0.156·16-s + 0.648·17-s + 0.186·18-s − 0.215·19-s − 1.03·20-s + 0.550·21-s + 0.417·23-s − 0.545·24-s + 1.27·25-s − 0.907·26-s + 0.192·27-s − 0.654·28-s − 0.147·29-s + 0.487·30-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: 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 2.2538261922.253826192
L(12)L(\frac12) ≈\approx 2.2538261922.253826192
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 1−0.792T+2T2 1 - 0.792T + 2T^{2}
5 1−3.37T+5T2 1 - 3.37T + 5T^{2}
7 1−2.52T+7T2 1 - 2.52T + 7T^{2}
13 1+5.84T+13T2 1 + 5.84T + 13T^{2}
17 1−2.67T+17T2 1 - 2.67T + 17T^{2}
19 1+0.939T+19T2 1 + 0.939T + 19T^{2}
23 1−2T+23T2 1 - 2T + 23T^{2}
29 1+0.792T+29T2 1 + 0.792T + 29T^{2}
31 1−1.62T+31T2 1 - 1.62T + 31T^{2}
37 1−5T+37T2 1 - 5T + 37T^{2}
41 1+10.8T+41T2 1 + 10.8T + 41T^{2}
43 1+6.63T+43T2 1 + 6.63T + 43T^{2}
47 1+12.7T+47T2 1 + 12.7T + 47T^{2}
53 1+4.11T+53T2 1 + 4.11T + 53T^{2}
59 1+6T+59T2 1 + 6T + 59T^{2}
61 1−5.98T+61T2 1 - 5.98T + 61T^{2}
67 1+1.11T+67T2 1 + 1.11T + 67T^{2}
71 1+10.7T+71T2 1 + 10.7T + 71T^{2}
73 1−9.15T+73T2 1 - 9.15T + 73T^{2}
79 1−4.10T+79T2 1 - 4.10T + 79T^{2}
83 1+1.87T+83T2 1 + 1.87T + 83T^{2}
89 1+0.627T+89T2 1 + 0.627T + 89T^{2}
97 1−10.4T+97T2 1 - 10.4T + 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

−11.62050949706971693175784826730, −10.08291896927709444050110178215, −9.734244861019423555355439770966, −8.752974647539106053582913938070, −7.79694174501864191774960110430, −6.44663193842469751787444206692, −5.20396291391023309423150626190, −4.76468449858083751139902884752, −3.08416565161952874084754534841, −1.83411454730264611735408611709, 1.83411454730264611735408611709, 3.08416565161952874084754534841, 4.76468449858083751139902884752, 5.20396291391023309423150626190, 6.44663193842469751787444206692, 7.79694174501864191774960110430, 8.752974647539106053582913938070, 9.734244861019423555355439770966, 10.08291896927709444050110178215, 11.62050949706971693175784826730

Graph of the ZZ-function along the critical line