Properties

Label 2-1575-1.1-c3-0-102
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.73·2-s + 5.94·4-s − 7·7-s + 7.67·8-s + 36.3·11-s + 75.8·13-s + 26.1·14-s − 76.2·16-s − 31.6·17-s + 25.1·19-s − 135.·22-s − 212.·23-s − 283.·26-s − 41.6·28-s + 235.·29-s − 270.·31-s + 223.·32-s + 118.·34-s − 362.·37-s − 93.9·38-s − 132.·41-s + 5.13·43-s + 216.·44-s + 792.·46-s − 216.·47-s + 49·49-s + 451.·52-s + ⋯
L(s)  = 1  − 1.32·2-s + 0.743·4-s − 0.377·7-s + 0.338·8-s + 0.996·11-s + 1.61·13-s + 0.499·14-s − 1.19·16-s − 0.451·17-s + 0.303·19-s − 1.31·22-s − 1.92·23-s − 2.13·26-s − 0.280·28-s + 1.50·29-s − 1.56·31-s + 1.23·32-s + 0.596·34-s − 1.60·37-s − 0.400·38-s − 0.505·41-s + 0.0182·43-s + 0.740·44-s + 2.54·46-s − 0.672·47-s + 0.142·49-s + 1.20·52-s + ⋯

Functional equation

Λ(s)=(1575s/2ΓC(s)L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1575s/2ΓC(s+3/2)L(s)=(Λ(1s)\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    =    325273^{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(ps)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+3.73T+8T2 1 + 3.73T + 8T^{2}
11 136.3T+1.33e3T2 1 - 36.3T + 1.33e3T^{2}
13 175.8T+2.19e3T2 1 - 75.8T + 2.19e3T^{2}
17 1+31.6T+4.91e3T2 1 + 31.6T + 4.91e3T^{2}
19 125.1T+6.85e3T2 1 - 25.1T + 6.85e3T^{2}
23 1+212.T+1.21e4T2 1 + 212.T + 1.21e4T^{2}
29 1235.T+2.43e4T2 1 - 235.T + 2.43e4T^{2}
31 1+270.T+2.97e4T2 1 + 270.T + 2.97e4T^{2}
37 1+362.T+5.06e4T2 1 + 362.T + 5.06e4T^{2}
41 1+132.T+6.89e4T2 1 + 132.T + 6.89e4T^{2}
43 15.13T+7.95e4T2 1 - 5.13T + 7.95e4T^{2}
47 1+216.T+1.03e5T2 1 + 216.T + 1.03e5T^{2}
53 1455.T+1.48e5T2 1 - 455.T + 1.48e5T^{2}
59 1689.T+2.05e5T2 1 - 689.T + 2.05e5T^{2}
61 1130.T+2.26e5T2 1 - 130.T + 2.26e5T^{2}
67 1+633.T+3.00e5T2 1 + 633.T + 3.00e5T^{2}
71 11.06e3T+3.57e5T2 1 - 1.06e3T + 3.57e5T^{2}
73 1+1.00e3T+3.89e5T2 1 + 1.00e3T + 3.89e5T^{2}
79 1+381.T+4.93e5T2 1 + 381.T + 4.93e5T^{2}
83 1+48.5T+5.71e5T2 1 + 48.5T + 5.71e5T^{2}
89 1+53.5T+7.04e5T2 1 + 53.5T + 7.04e5T^{2}
97 1+968.T+9.12e5T2 1 + 968.T + 9.12e5T^{2}
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   L(s)=p j=12(1αj,pps)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

−8.600482070559689503932093754458, −8.294818317983626520707626125187, −7.09742558722274511898808789670, −6.53927626910810476034512440344, −5.63046171702231332212903147999, −4.22798165998469826770686055857, −3.53593801720265316930473974944, −1.98062764744021305272744132256, −1.16416871364992294215086809502, 0, 1.16416871364992294215086809502, 1.98062764744021305272744132256, 3.53593801720265316930473974944, 4.22798165998469826770686055857, 5.63046171702231332212903147999, 6.53927626910810476034512440344, 7.09742558722274511898808789670, 8.294818317983626520707626125187, 8.600482070559689503932093754458

Graph of the ZZ-function along the critical line