↑

Properties

Label 2-175-1.1-c3-0-27
Degree 22
Conductor 175175
Sign −1-1
Analytic cond. 10.325310.3253
Root an. cond. 3.213303.21330
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  + 1.67·2-s + 2.49·3-s − 5.19·4-s + 4.17·6-s − 7·7-s − 22.1·8-s − 20.7·9-s − 57.5·11-s − 12.9·12-s + 45.5·13-s − 11.7·14-s + 4.52·16-s − 92.0·17-s − 34.8·18-s + 125.·19-s − 17.4·21-s − 96.4·22-s − 158.·23-s − 55.1·24-s + 76.2·26-s − 119.·27-s + 36.3·28-s − 40.1·29-s + 49.5·31-s + 184.·32-s − 143.·33-s − 154.·34-s + ⋯
L(s)  = 1  + 0.592·2-s + 0.479·3-s − 0.649·4-s + 0.284·6-s − 0.377·7-s − 0.976·8-s − 0.769·9-s − 1.57·11-s − 0.311·12-s + 0.971·13-s − 0.223·14-s + 0.0707·16-s − 1.31·17-s − 0.455·18-s + 1.51·19-s − 0.181·21-s − 0.934·22-s − 1.43·23-s − 0.468·24-s + 0.575·26-s − 0.849·27-s + 0.245·28-s − 0.257·29-s + 0.287·31-s + 1.01·32-s − 0.757·33-s − 0.777·34-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −1-1
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 175, ( :3/2), −1)(2,\ 175,\ (\ :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
7 1+7T 1 + 7T
good2 1−1.67T+8T2 1 - 1.67T + 8T^{2}
3 1−2.49T+27T2 1 - 2.49T + 27T^{2}
11 1+57.5T+1.33e3T2 1 + 57.5T + 1.33e3T^{2}
13 1−45.5T+2.19e3T2 1 - 45.5T + 2.19e3T^{2}
17 1+92.0T+4.91e3T2 1 + 92.0T + 4.91e3T^{2}
19 1−125.T+6.85e3T2 1 - 125.T + 6.85e3T^{2}
23 1+158.T+1.21e4T2 1 + 158.T + 1.21e4T^{2}
29 1+40.1T+2.43e4T2 1 + 40.1T + 2.43e4T^{2}
31 1−49.5T+2.97e4T2 1 - 49.5T + 2.97e4T^{2}
37 1+231.T+5.06e4T2 1 + 231.T + 5.06e4T^{2}
41 1−169.T+6.89e4T2 1 - 169.T + 6.89e4T^{2}
43 1−147.T+7.95e4T2 1 - 147.T + 7.95e4T^{2}
47 1−67.0T+1.03e5T2 1 - 67.0T + 1.03e5T^{2}
53 1+268.T+1.48e5T2 1 + 268.T + 1.48e5T^{2}
59 1+240.T+2.05e5T2 1 + 240.T + 2.05e5T^{2}
61 1−90.4T+2.26e5T2 1 - 90.4T + 2.26e5T^{2}
67 1−406.T+3.00e5T2 1 - 406.T + 3.00e5T^{2}
71 1−330.T+3.57e5T2 1 - 330.T + 3.57e5T^{2}
73 1+546.T+3.89e5T2 1 + 546.T + 3.89e5T^{2}
79 1+25.3T+4.93e5T2 1 + 25.3T + 4.93e5T^{2}
83 1+376.T+5.71e5T2 1 + 376.T + 5.71e5T^{2}
89 1−1.02e3T+7.04e5T2 1 - 1.02e3T + 7.04e5T^{2}
97 1−942.T+9.12e5T2 1 - 942.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

−11.95727913560625512753860126665, −10.80952916947205130735145169703, −9.602708671841040568857858336486, −8.675240192038183125173313709889, −7.81176034315380149747627902391, −6.09335020160972566054338377833, −5.19556475339900537700143597115, −3.76859841798221154017980803385, −2.67380837599508768975615599995, 0, 2.67380837599508768975615599995, 3.76859841798221154017980803385, 5.19556475339900537700143597115, 6.09335020160972566054338377833, 7.81176034315380149747627902391, 8.675240192038183125173313709889, 9.602708671841040568857858336486, 10.80952916947205130735145169703, 11.95727913560625512753860126665

Graph of the ZZ-function along the critical line