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Properties

Label 2-504-168.83-c0-0-0
Degree 22
Conductor 504504
Sign 0.985−0.169i0.985 - 0.169i
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + i·7-s + (0.707 − 0.707i)8-s + 1.41i·11-s + (0.707 − 0.707i)14-s − 1.00·16-s + (1.00 − 1.00i)22-s + 1.41·23-s + 25-s − 1.00·28-s − 1.41·29-s + (0.707 + 0.707i)32-s − 2i·37-s − 1.41·44-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + i·7-s + (0.707 − 0.707i)8-s + 1.41i·11-s + (0.707 − 0.707i)14-s − 1.00·16-s + (1.00 − 1.00i)22-s + 1.41·23-s + 25-s − 1.00·28-s − 1.41·29-s + (0.707 + 0.707i)32-s − 2i·37-s − 1.41·44-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=((0.985−0.169i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s) L(s)=((0.985−0.169i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: 0.985−0.169i0.985 - 0.169i
Analytic conductor: 0.2515280.251528
Root analytic conductor: 0.5015260.501526
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ504(251,⋅)\chi_{504} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 504, ( :0), 0.985−0.169i)(2,\ 504,\ (\ :0),\ 0.985 - 0.169i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.62370915260.6237091526
L(12)L(\frac12) ≈\approx 0.62370915260.6237091526
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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
3 1 1
7 1−iT 1 - iT
good5 1−T2 1 - T^{2}
11 1−1.41iT−T2 1 - 1.41iT - T^{2}
13 1+T2 1 + T^{2}
17 1+T2 1 + T^{2}
19 1−T2 1 - T^{2}
23 1−1.41T+T2 1 - 1.41T + T^{2}
29 1+1.41T+T2 1 + 1.41T + T^{2}
31 1+T2 1 + T^{2}
37 1+2iT−T2 1 + 2iT - T^{2}
41 1+T2 1 + T^{2}
43 1+T2 1 + T^{2}
47 1−T2 1 - T^{2}
53 1−1.41T+T2 1 - 1.41T + T^{2}
59 1+T2 1 + T^{2}
61 1+T2 1 + T^{2}
67 1+2T+T2 1 + 2T + T^{2}
71 1+1.41T+T2 1 + 1.41T + T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - 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

−11.09539624586277659311996316471, −10.27870536891162848238623005228, −9.197864045091260941480818990380, −8.941340081773811902399468238072, −7.60283446408897253793407365398, −6.93857146686148557311978969589, −5.43651881898298964296186649520, −4.29975263480637199568527049795, −2.90218956359490018053606238543, −1.84942642359722108302307649016, 1.11445972173035772105677404368, 3.19400757901157965243347857499, 4.64821740335521403509026352187, 5.71774425844654779051650892619, 6.71996813128602617644230951416, 7.46626462789162949627407698407, 8.461477255022252256962400027852, 9.131272453075463139142778115489, 10.24724136289547206026891892779, 10.88108714208982667795883332981

Graph of the ZZ-function along the critical line