↑

Properties

Label 2-1575-1.1-c3-0-103
Degree 22
Conductor 15751575
Sign −1-1
Analytic cond. 92.928092.9280
Root an. cond. 9.639919.63991
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  + 0.504·2-s − 7.74·4-s + 7·7-s − 7.94·8-s − 54.8·11-s + 16.0·13-s + 3.53·14-s + 57.9·16-s + 0.422·17-s + 127.·19-s − 27.7·22-s − 51.1·23-s + 8.08·26-s − 54.2·28-s − 41.4·29-s + 192.·31-s + 92.8·32-s + 0.213·34-s − 189.·37-s + 64.3·38-s + 76.3·41-s + 294.·43-s + 425.·44-s − 25.7·46-s − 540.·47-s + 49·49-s − 123.·52-s + ⋯
L(s)  = 1  + 0.178·2-s − 0.968·4-s + 0.377·7-s − 0.351·8-s − 1.50·11-s + 0.341·13-s + 0.0674·14-s + 0.905·16-s + 0.00602·17-s + 1.53·19-s − 0.268·22-s − 0.463·23-s + 0.0609·26-s − 0.365·28-s − 0.265·29-s + 1.11·31-s + 0.512·32-s + 0.00107·34-s − 0.840·37-s + 0.274·38-s + 0.290·41-s + 1.04·43-s + 1.45·44-s − 0.0826·46-s − 1.67·47-s + 0.142·49-s − 0.330·52-s + ⋯

Functional equation

Λ(s)=(1575s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1575s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15751575    =    32⋅52⋅73^{2} \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 92.928092.9280
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1575, ( :3/2), −1)(2,\ 1575,\ (\ :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)
bad3 1 1
5 1 1
7 1−7T 1 - 7T
good2 1−0.504T+8T2 1 - 0.504T + 8T^{2}
11 1+54.8T+1.33e3T2 1 + 54.8T + 1.33e3T^{2}
13 1−16.0T+2.19e3T2 1 - 16.0T + 2.19e3T^{2}
17 1−0.422T+4.91e3T2 1 - 0.422T + 4.91e3T^{2}
19 1−127.T+6.85e3T2 1 - 127.T + 6.85e3T^{2}
23 1+51.1T+1.21e4T2 1 + 51.1T + 1.21e4T^{2}
29 1+41.4T+2.43e4T2 1 + 41.4T + 2.43e4T^{2}
31 1−192.T+2.97e4T2 1 - 192.T + 2.97e4T^{2}
37 1+189.T+5.06e4T2 1 + 189.T + 5.06e4T^{2}
41 1−76.3T+6.89e4T2 1 - 76.3T + 6.89e4T^{2}
43 1−294.T+7.95e4T2 1 - 294.T + 7.95e4T^{2}
47 1+540.T+1.03e5T2 1 + 540.T + 1.03e5T^{2}
53 1−661.T+1.48e5T2 1 - 661.T + 1.48e5T^{2}
59 1+410.T+2.05e5T2 1 + 410.T + 2.05e5T^{2}
61 1−46.0T+2.26e5T2 1 - 46.0T + 2.26e5T^{2}
67 1−10.4T+3.00e5T2 1 - 10.4T + 3.00e5T^{2}
71 1−491.T+3.57e5T2 1 - 491.T + 3.57e5T^{2}
73 1+814.T+3.89e5T2 1 + 814.T + 3.89e5T^{2}
79 1+858.T+4.93e5T2 1 + 858.T + 4.93e5T^{2}
83 1−1.05e3T+5.71e5T2 1 - 1.05e3T + 5.71e5T^{2}
89 1+341.T+7.04e5T2 1 + 341.T + 7.04e5T^{2}
97 1+1.41e3T+9.12e5T2 1 + 1.41e3T + 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

−8.538877691483014295633354723248, −7.995127227814632421923625035794, −7.25305955713484787817781560238, −5.90348465202619142901548779861, −5.28679625616914132438790769323, −4.60361337773776209181337607797, −3.55528783844966165002727701114, −2.63785556257609362022225206533, −1.16458108812078835901304952419, 0, 1.16458108812078835901304952419, 2.63785556257609362022225206533, 3.55528783844966165002727701114, 4.60361337773776209181337607797, 5.28679625616914132438790769323, 5.90348465202619142901548779861, 7.25305955713484787817781560238, 7.995127227814632421923625035794, 8.538877691483014295633354723248

Graph of the ZZ-function along the critical line