Properties

Label 2-3825-1.1-c1-0-97
Degree 22
Conductor 38253825
Sign 1-1
Analytic cond. 30.542730.5427
Root an. cond. 5.526555.52655
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.523·2-s − 1.72·4-s + 3.20·7-s − 1.95·8-s − 5.20·11-s − 2·13-s + 1.67·14-s + 2.42·16-s − 17-s + 7.20·19-s − 2.72·22-s − 1.04·26-s − 5.52·28-s + 0.249·29-s + 6.40·31-s + 5.17·32-s − 0.523·34-s − 2.24·37-s + 3.77·38-s − 10.6·41-s − 4.09·43-s + 8.97·44-s − 11.7·47-s + 3.24·49-s + 3.45·52-s − 1.84·53-s − 6.24·56-s + ⋯
L(s)  = 1  + 0.370·2-s − 0.862·4-s + 1.21·7-s − 0.690·8-s − 1.56·11-s − 0.554·13-s + 0.448·14-s + 0.607·16-s − 0.242·17-s + 1.65·19-s − 0.581·22-s − 0.205·26-s − 1.04·28-s + 0.0463·29-s + 1.15·31-s + 0.915·32-s − 0.0898·34-s − 0.369·37-s + 0.612·38-s − 1.66·41-s − 0.624·43-s + 1.35·44-s − 1.70·47-s + 0.464·49-s + 0.478·52-s − 0.253·53-s − 0.835·56-s + ⋯

Functional equation

Λ(s)=(3825s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(3825s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38253825    =    3252173^{2} \cdot 5^{2} \cdot 17
Sign: 1-1
Analytic conductor: 30.542730.5427
Root analytic conductor: 5.526555.52655
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 3825, ( :1/2), 1)(2,\ 3825,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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
17 1+T 1 + T
good2 10.523T+2T2 1 - 0.523T + 2T^{2}
7 13.20T+7T2 1 - 3.20T + 7T^{2}
11 1+5.20T+11T2 1 + 5.20T + 11T^{2}
13 1+2T+13T2 1 + 2T + 13T^{2}
19 17.20T+19T2 1 - 7.20T + 19T^{2}
23 1+23T2 1 + 23T^{2}
29 10.249T+29T2 1 - 0.249T + 29T^{2}
31 16.40T+31T2 1 - 6.40T + 31T^{2}
37 1+2.24T+37T2 1 + 2.24T + 37T^{2}
41 1+10.6T+41T2 1 + 10.6T + 41T^{2}
43 1+4.09T+43T2 1 + 4.09T + 43T^{2}
47 1+11.7T+47T2 1 + 11.7T + 47T^{2}
53 1+1.84T+53T2 1 + 1.84T + 53T^{2}
59 110.4T+59T2 1 - 10.4T + 59T^{2}
61 1+10.4T+61T2 1 + 10.4T + 61T^{2}
67 1+10.3T+67T2 1 + 10.3T + 67T^{2}
71 16.49T+71T2 1 - 6.49T + 71T^{2}
73 1+3.84T+73T2 1 + 3.84T + 73T^{2}
79 18T+79T2 1 - 8T + 79T^{2}
83 18.80T+83T2 1 - 8.80T + 83T^{2}
89 1+8.30T+89T2 1 + 8.30T + 89T^{2}
97 14.49T+97T2 1 - 4.49T + 97T^{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

−8.044437080496471345196971314710, −7.69255023408653142314211767764, −6.62407693923615115514087328210, −5.42131649431182377395067522679, −5.07962152504064313832627078401, −4.62767744872691362034956928828, −3.43185779021311095998451336384, −2.66577050232533439714139098675, −1.41058966609962394730005058603, 0, 1.41058966609962394730005058603, 2.66577050232533439714139098675, 3.43185779021311095998451336384, 4.62767744872691362034956928828, 5.07962152504064313832627078401, 5.42131649431182377395067522679, 6.62407693923615115514087328210, 7.69255023408653142314211767764, 8.044437080496471345196971314710

Graph of the ZZ-function along the critical line