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Properties

Label 2-504-21.11-c0-0-0
Degree 22
Conductor 504504
Sign 0.475−0.879i0.475 - 0.879i
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  + (−1.22 + 0.707i)5-s + (0.5 + 0.866i)7-s + (1.22 + 0.707i)11-s − 13-s + (0.5 + 0.866i)19-s + (0.499 − 0.866i)25-s + (0.5 − 0.866i)31-s + (−1.22 − 0.707i)35-s + (−0.5 − 0.866i)37-s − 1.41i·41-s − 43-s + (1.22 − 0.707i)47-s + (−0.499 + 0.866i)49-s − 2·55-s + (1.22 − 0.707i)65-s + ⋯
L(s)  = 1  + (−1.22 + 0.707i)5-s + (0.5 + 0.866i)7-s + (1.22 + 0.707i)11-s − 13-s + (0.5 + 0.866i)19-s + (0.499 − 0.866i)25-s + (0.5 − 0.866i)31-s + (−1.22 − 0.707i)35-s + (−0.5 − 0.866i)37-s − 1.41i·41-s − 43-s + (1.22 − 0.707i)47-s + (−0.499 + 0.866i)49-s − 2·55-s + (1.22 − 0.707i)65-s + ⋯

Functional equation

Λ(s)=(504s/2 ΓC(s) L(s)=((0.475−0.879i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(504s/2 ΓC(s) L(s)=((0.475−0.879i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 504504    =    23⋅32⋅72^{3} \cdot 3^{2} \cdot 7
Sign: 0.475−0.879i0.475 - 0.879i
Analytic conductor: 0.2515280.251528
Root analytic conductor: 0.5015260.501526
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ504(305,⋅)\chi_{504} (305, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 504, ( :0), 0.475−0.879i)(2,\ 504,\ (\ :0),\ 0.475 - 0.879i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.76202682930.7620268293
L(12)L(\frac12) ≈\approx 0.76202682930.7620268293
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 1
3 1 1
7 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good5 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
11 1+(−1.22−0.707i)T+(0.5+0.866i)T2 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}
13 1+T+T2 1 + T + T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
41 1+1.41iT−T2 1 + 1.41iT - T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+(−1.22+0.707i)T+(0.5−0.866i)T2 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
67 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
71 1−1.41iT−T2 1 - 1.41iT - T^{2}
73 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
83 1+1.41iT−T2 1 + 1.41iT - T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)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

−11.66011663630116451726390585150, −10.45522636167720713992185071297, −9.499968631698451319253150087457, −8.549527855552450035468459675980, −7.55917077529776587537660342935, −6.97827839854047070715630623545, −5.70151990847289172330832249379, −4.47449960971579833236472073341, −3.54170474871652206340908868216, −2.13280923285315758235947114459, 1.08160545305569271639776234426, 3.27592709468854609113046103458, 4.33329557414190601727818274402, 4.98073153966592754997594122412, 6.60821740181229889229126340408, 7.44815481835568423864427563048, 8.267986284762705842757313145066, 9.051547773268775539995843567603, 10.11724510002133269095910640477, 11.28622041011387475511905024813

Graph of the ZZ-function along the critical line