↑

Properties

Label 2-275-1.1-c3-0-16
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  − 4.78·2-s − 5.99·3-s + 14.8·4-s + 28.6·6-s − 11.9·7-s − 32.7·8-s + 8.91·9-s − 11·11-s − 89.0·12-s − 22.4·13-s + 57.1·14-s + 37.8·16-s + 131.·17-s − 42.5·18-s + 99.0·19-s + 71.6·21-s + 52.5·22-s + 0.206·23-s + 196.·24-s + 107.·26-s + 108.·27-s − 177.·28-s − 163.·29-s + 217.·31-s + 81.3·32-s + 65.9·33-s − 630.·34-s + ⋯
L(s)  = 1  − 1.69·2-s − 1.15·3-s + 1.85·4-s + 1.94·6-s − 0.645·7-s − 1.44·8-s + 0.330·9-s − 0.301·11-s − 2.14·12-s − 0.478·13-s + 1.09·14-s + 0.591·16-s + 1.88·17-s − 0.557·18-s + 1.19·19-s + 0.745·21-s + 0.509·22-s + 0.00187·23-s + 1.67·24-s + 0.808·26-s + 0.772·27-s − 1.19·28-s − 1.04·29-s + 1.25·31-s + 0.449·32-s + 0.347·33-s − 3.18·34-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+4.78T+8T2 1 + 4.78T + 8T^{2}
3 1+5.99T+27T2 1 + 5.99T + 27T^{2}
7 1+11.9T+343T2 1 + 11.9T + 343T^{2}
13 1+22.4T+2.19e3T2 1 + 22.4T + 2.19e3T^{2}
17 1−131.T+4.91e3T2 1 - 131.T + 4.91e3T^{2}
19 1−99.0T+6.85e3T2 1 - 99.0T + 6.85e3T^{2}
23 1−0.206T+1.21e4T2 1 - 0.206T + 1.21e4T^{2}
29 1+163.T+2.43e4T2 1 + 163.T + 2.43e4T^{2}
31 1−217.T+2.97e4T2 1 - 217.T + 2.97e4T^{2}
37 1+17.8T+5.06e4T2 1 + 17.8T + 5.06e4T^{2}
41 1+32.3T+6.89e4T2 1 + 32.3T + 6.89e4T^{2}
43 1+490.T+7.95e4T2 1 + 490.T + 7.95e4T^{2}
47 1−518.T+1.03e5T2 1 - 518.T + 1.03e5T^{2}
53 1+110.T+1.48e5T2 1 + 110.T + 1.48e5T^{2}
59 1−242.T+2.05e5T2 1 - 242.T + 2.05e5T^{2}
61 1+713.T+2.26e5T2 1 + 713.T + 2.26e5T^{2}
67 1+571.T+3.00e5T2 1 + 571.T + 3.00e5T^{2}
71 1+113.T+3.57e5T2 1 + 113.T + 3.57e5T^{2}
73 1+767.T+3.89e5T2 1 + 767.T + 3.89e5T^{2}
79 1−470.T+4.93e5T2 1 - 470.T + 4.93e5T^{2}
83 1−1.15e3T+5.71e5T2 1 - 1.15e3T + 5.71e5T^{2}
89 1−719.T+7.04e5T2 1 - 719.T + 7.04e5T^{2}
97 1−510.T+9.12e5T2 1 - 510.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

−10.66223381062421851381717165748, −10.02873684117439981694233221342, −9.338317822406418731122198895795, −8.019774517873546920009177244123, −7.24917997419879368221081199437, −6.18161772006151311220534969411, −5.21150253514238299909884249974, −3.03899168649630070547656221715, −1.16079970326181627380577536905, 0, 1.16079970326181627380577536905, 3.03899168649630070547656221715, 5.21150253514238299909884249974, 6.18161772006151311220534969411, 7.24917997419879368221081199437, 8.019774517873546920009177244123, 9.338317822406418731122198895795, 10.02873684117439981694233221342, 10.66223381062421851381717165748

Graph of the ZZ-function along the critical line