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Properties

Label 2-1575-1.1-c3-0-46
Degree 22
Conductor 15751575
Sign 11
Analytic cond. 92.928092.9280
Root an. cond. 9.639919.63991
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 4-s − 7·7-s + 21·8-s + 60·11-s − 38·13-s + 21·14-s − 71·16-s + 84·17-s + 110·19-s − 180·22-s − 120·23-s + 114·26-s − 7·28-s + 162·29-s + 236·31-s + 45·32-s − 252·34-s + 376·37-s − 330·38-s − 126·41-s + 34·43-s + 60·44-s + 360·46-s + 6·47-s + 49·49-s − 38·52-s + ⋯
L(s)  = 1  − 1.06·2-s + 1/8·4-s − 0.377·7-s + 0.928·8-s + 1.64·11-s − 0.810·13-s + 0.400·14-s − 1.10·16-s + 1.19·17-s + 1.32·19-s − 1.74·22-s − 1.08·23-s + 0.859·26-s − 0.0472·28-s + 1.03·29-s + 1.36·31-s + 0.248·32-s − 1.27·34-s + 1.67·37-s − 1.40·38-s − 0.479·41-s + 0.120·43-s + 0.205·44-s + 1.15·46-s + 0.0186·47-s + 1/7·49-s − 0.101·52-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: 11
Analytic conductor: 92.928092.9280
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1575, ( :3/2), 1)(2,\ 1575,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.2594581451.259458145
L(12)L(\frac12) ≈\approx 1.2594581451.259458145
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+pT 1 + p T
good2 1+3T+p3T2 1 + 3 T + p^{3} T^{2}
11 1−60T+p3T2 1 - 60 T + p^{3} T^{2}
13 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
17 1−84T+p3T2 1 - 84 T + p^{3} T^{2}
19 1−110T+p3T2 1 - 110 T + p^{3} T^{2}
23 1+120T+p3T2 1 + 120 T + p^{3} T^{2}
29 1−162T+p3T2 1 - 162 T + p^{3} T^{2}
31 1−236T+p3T2 1 - 236 T + p^{3} T^{2}
37 1−376T+p3T2 1 - 376 T + p^{3} T^{2}
41 1+126T+p3T2 1 + 126 T + p^{3} T^{2}
43 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
47 1−6T+p3T2 1 - 6 T + p^{3} T^{2}
53 1+582T+p3T2 1 + 582 T + p^{3} T^{2}
59 1−492T+p3T2 1 - 492 T + p^{3} T^{2}
61 1+880T+p3T2 1 + 880 T + p^{3} T^{2}
67 1−826T+p3T2 1 - 826 T + p^{3} T^{2}
71 1+666T+p3T2 1 + 666 T + p^{3} T^{2}
73 1−826T+p3T2 1 - 826 T + p^{3} T^{2}
79 1+592T+p3T2 1 + 592 T + p^{3} T^{2}
83 1+792T+p3T2 1 + 792 T + p^{3} T^{2}
89 1−1002T+p3T2 1 - 1002 T + p^{3} T^{2}
97 1+1442T+p3T2 1 + 1442 T + p^{3} T^{2}
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   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

−9.275199666107111971597503930626, −8.262671905598554008194260425412, −7.69937519011113004098244554931, −6.83276582215055888044447485549, −6.02270053679299264650567972482, −4.84044872568776255138190729034, −3.99165220698615545959163756334, −2.88500847997515113332739930679, −1.45194209929227002936861618932, −0.71674928355056388130186062589, 0.71674928355056388130186062589, 1.45194209929227002936861618932, 2.88500847997515113332739930679, 3.99165220698615545959163756334, 4.84044872568776255138190729034, 6.02270053679299264650567972482, 6.83276582215055888044447485549, 7.69937519011113004098244554931, 8.262671905598554008194260425412, 9.275199666107111971597503930626

Graph of the ZZ-function along the critical line