↑

Properties

Label 2-275-1.1-c1-0-15
Degree 22
Conductor 275275
Sign −1-1
Analytic cond. 2.195882.19588
Root an. cond. 1.481851.48185
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.618·2-s − 0.381·3-s − 1.61·4-s − 0.236·6-s − 3.85·7-s − 2.23·8-s − 2.85·9-s + 11-s + 0.618·12-s − 1.76·13-s − 2.38·14-s + 1.85·16-s − 1.61·17-s − 1.76·18-s + 6.70·19-s + 1.47·21-s + 0.618·22-s − 7.09·23-s + 0.854·24-s − 1.09·26-s + 2.23·27-s + 6.23·28-s − 3.61·29-s − 3·31-s + 5.61·32-s − 0.381·33-s − 1.00·34-s + ⋯
L(s)  = 1  + 0.437·2-s − 0.220·3-s − 0.809·4-s − 0.0963·6-s − 1.45·7-s − 0.790·8-s − 0.951·9-s + 0.301·11-s + 0.178·12-s − 0.489·13-s − 0.636·14-s + 0.463·16-s − 0.392·17-s − 0.415·18-s + 1.53·19-s + 0.321·21-s + 0.131·22-s − 1.47·23-s + 0.174·24-s − 0.213·26-s + 0.430·27-s + 1.17·28-s − 0.671·29-s − 0.538·31-s + 0.993·32-s − 0.0664·33-s − 0.171·34-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: −1-1
Analytic conductor: 2.195882.19588
Root analytic conductor: 1.481851.48185
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 275, ( :1/2), −1)(2,\ 275,\ (\ :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)
bad5 1 1
11 1−T 1 - T
good2 1−0.618T+2T2 1 - 0.618T + 2T^{2}
3 1+0.381T+3T2 1 + 0.381T + 3T^{2}
7 1+3.85T+7T2 1 + 3.85T + 7T^{2}
13 1+1.76T+13T2 1 + 1.76T + 13T^{2}
17 1+1.61T+17T2 1 + 1.61T + 17T^{2}
19 1−6.70T+19T2 1 - 6.70T + 19T^{2}
23 1+7.09T+23T2 1 + 7.09T + 23T^{2}
29 1+3.61T+29T2 1 + 3.61T + 29T^{2}
31 1+3T+31T2 1 + 3T + 31T^{2}
37 1+5.76T+37T2 1 + 5.76T + 37T^{2}
41 1+3T+41T2 1 + 3T + 41T^{2}
43 1−6T+43T2 1 - 6T + 43T^{2}
47 1−5.94T+47T2 1 - 5.94T + 47T^{2}
53 1−6.32T+53T2 1 - 6.32T + 53T^{2}
59 1−9.47T+59T2 1 - 9.47T + 59T^{2}
61 1+11.0T+61T2 1 + 11.0T + 61T^{2}
67 1+8T+67T2 1 + 8T + 67T^{2}
71 1+14.1T+71T2 1 + 14.1T + 71T^{2}
73 1+12.6T+73T2 1 + 12.6T + 73T^{2}
79 1+0.854T+79T2 1 + 0.854T + 79T^{2}
83 1−16.8T+83T2 1 - 16.8T + 83T^{2}
89 1+18.0T+89T2 1 + 18.0T + 89T^{2}
97 1−0.618T+97T2 1 - 0.618T + 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

−11.80555197965088696893994809316, −10.31706666720708864964347944408, −9.463444875934616351750339132614, −8.802682250912539586751346681205, −7.38623110876223889945537992955, −6.09098378861924905440480640353, −5.42670884232244898599868521037, −3.95430665306330740889269912878, −2.96886712006022722229163981006, 0, 2.96886712006022722229163981006, 3.95430665306330740889269912878, 5.42670884232244898599868521037, 6.09098378861924905440480640353, 7.38623110876223889945537992955, 8.802682250912539586751346681205, 9.463444875934616351750339132614, 10.31706666720708864964347944408, 11.80555197965088696893994809316

Graph of the ZZ-function along the critical line