↑

Properties

Label 2-275-1.1-c3-0-38
Degree 22
Conductor 275275
Sign −1-1
Analytic cond. 16.225516.2255
Root an. cond. 4.028094.02809
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.28·2-s − 2.58·3-s − 2.79·4-s − 5.90·6-s + 27.1·7-s − 24.6·8-s − 20.3·9-s + 11·11-s + 7.22·12-s + 9.09·13-s + 61.9·14-s − 33.8·16-s − 91.1·17-s − 46.3·18-s − 80.9·19-s − 70.2·21-s + 25.0·22-s − 208.·23-s + 63.7·24-s + 20.7·26-s + 122.·27-s − 75.8·28-s + 136.·29-s − 213.·31-s + 119.·32-s − 28.4·33-s − 207.·34-s + ⋯
L(s)  = 1  + 0.806·2-s − 0.497·3-s − 0.349·4-s − 0.401·6-s + 1.46·7-s − 1.08·8-s − 0.752·9-s + 0.301·11-s + 0.173·12-s + 0.194·13-s + 1.18·14-s − 0.528·16-s − 1.30·17-s − 0.606·18-s − 0.977·19-s − 0.729·21-s + 0.243·22-s − 1.88·23-s + 0.541·24-s + 0.156·26-s + 0.872·27-s − 0.511·28-s + 0.872·29-s − 1.23·31-s + 0.661·32-s − 0.150·33-s − 1.04·34-s + ⋯

Functional equation

Λ(s)=(275s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(275s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 275275    =    52⋅115^{2} \cdot 11
Sign: −1-1
Analytic conductor: 16.225516.2255
Root analytic conductor: 4.028094.02809
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 275, ( :3/2), −1)(2,\ 275,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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 1−11T 1 - 11T
good2 1−2.28T+8T2 1 - 2.28T + 8T^{2}
3 1+2.58T+27T2 1 + 2.58T + 27T^{2}
7 1−27.1T+343T2 1 - 27.1T + 343T^{2}
13 1−9.09T+2.19e3T2 1 - 9.09T + 2.19e3T^{2}
17 1+91.1T+4.91e3T2 1 + 91.1T + 4.91e3T^{2}
19 1+80.9T+6.85e3T2 1 + 80.9T + 6.85e3T^{2}
23 1+208.T+1.21e4T2 1 + 208.T + 1.21e4T^{2}
29 1−136.T+2.43e4T2 1 - 136.T + 2.43e4T^{2}
31 1+213.T+2.97e4T2 1 + 213.T + 2.97e4T^{2}
37 1+351.T+5.06e4T2 1 + 351.T + 5.06e4T^{2}
41 1+1.21T+6.89e4T2 1 + 1.21T + 6.89e4T^{2}
43 1+231.T+7.95e4T2 1 + 231.T + 7.95e4T^{2}
47 1+283.T+1.03e5T2 1 + 283.T + 1.03e5T^{2}
53 1−238.T+1.48e5T2 1 - 238.T + 1.48e5T^{2}
59 1−740.T+2.05e5T2 1 - 740.T + 2.05e5T^{2}
61 1−446.T+2.26e5T2 1 - 446.T + 2.26e5T^{2}
67 1+56.2T+3.00e5T2 1 + 56.2T + 3.00e5T^{2}
71 1−684.T+3.57e5T2 1 - 684.T + 3.57e5T^{2}
73 1+428.T+3.89e5T2 1 + 428.T + 3.89e5T^{2}
79 1+1.26e3T+4.93e5T2 1 + 1.26e3T + 4.93e5T^{2}
83 1+147.T+5.71e5T2 1 + 147.T + 5.71e5T^{2}
89 1−305.T+7.04e5T2 1 - 305.T + 7.04e5T^{2}
97 1+205.T+9.12e5T2 1 + 205.T + 9.12e5T^{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

−11.35833461559363289510649610378, −10.28565605243334638181519282200, −8.710979010477534752362927560307, −8.362795277592424680327795198242, −6.65246470465242875657853315457, −5.64443955631968414901410101741, −4.79219684052546232924181651648, −3.88695154171140260814616870886, −2.07944707267507891669780160271, 0, 2.07944707267507891669780160271, 3.88695154171140260814616870886, 4.79219684052546232924181651648, 5.64443955631968414901410101741, 6.65246470465242875657853315457, 8.362795277592424680327795198242, 8.710979010477534752362927560307, 10.28565605243334638181519282200, 11.35833461559363289510649610378

Graph of the ZZ-function along the critical line