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Properties

Label 2-504-56.13-c0-0-1
Degree 22
Conductor 504504
Sign ii
Analytic cond. 0.2515280.251528
Root an. cond. 0.5015260.501526
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 4-s + 7-s + i·8-s − 2i·11-s − i·14-s + 16-s − 2·22-s − 25-s − 28-s + 2i·29-s − i·32-s + 2i·44-s + 49-s + i·50-s + ⋯
L(s)  = 1  − i·2-s − 4-s + 7-s + i·8-s − 2i·11-s − i·14-s + 16-s − 2·22-s − 25-s − 28-s + 2i·29-s − i·32-s + 2i·44-s + 49-s + i·50-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: ii
Analytic conductor: 0.2515280.251528
Root analytic conductor: 0.5015260.501526
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ504(181,⋅)\chi_{504} (181, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 504, ( :0), i)(2,\ 504,\ (\ :0),\ i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.85950649730.8595064973
L(12)L(\frac12) ≈\approx 0.85950649730.8595064973
L(1)L(1) 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)
bad2 1+iT 1 + iT
3 1 1
7 1−T 1 - T
good5 1+T2 1 + T^{2}
11 1+2iT−T2 1 + 2iT - T^{2}
13 1+T2 1 + T^{2}
17 1−T2 1 - T^{2}
19 1+T2 1 + T^{2}
23 1+T2 1 + T^{2}
29 1−2iT−T2 1 - 2iT - T^{2}
31 1−T2 1 - T^{2}
37 1−T2 1 - T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1−T2 1 - T^{2}
53 1−2iT−T2 1 - 2iT - T^{2}
59 1+T2 1 + T^{2}
61 1+T2 1 + T^{2}
67 1−T2 1 - T^{2}
71 1+T2 1 + T^{2}
73 1−T2 1 - T^{2}
79 1+2T+T2 1 + 2T + T^{2}
83 1+T2 1 + T^{2}
89 1−T2 1 - T^{2}
97 1−T2 1 - 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

−11.02736288555510689890473445734, −10.35107207482332204488348574358, −9.027915968570364710771121702588, −8.534015845060712995438600939245, −7.61542465395263608247968716370, −5.95012340013087900626517670500, −5.13862652968003523422956471952, −3.92634248655435326341959254362, −2.89421827024905918171286179056, −1.34936159779161659946630610031, 1.98369331142917368553873576877, 4.11112593998298205424011014724, 4.75866191754171657278325285436, 5.79300872100055039029290891748, 6.97513040651102840211018154660, 7.67717943805798144423115023471, 8.402449026000533424865704733239, 9.627977813884985640099135376941, 10.07024174942785458389292631485, 11.47242509949491631579469741844

Graph of the ZZ-function along the critical line