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Properties

Label 2-483-23.22-c2-0-0
Degree 22
Conductor 483483
Sign −0.979−0.200i-0.979 - 0.200i
Analytic cond. 13.160713.1607
Root an. cond. 3.627783.62778
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.984·2-s + 1.73·3-s − 3.03·4-s − 2.95i·5-s − 1.70·6-s + 2.64i·7-s + 6.91·8-s + 2.99·9-s + 2.91i·10-s + 1.64i·11-s − 5.25·12-s − 20.6·13-s − 2.60i·14-s − 5.12i·15-s + 5.31·16-s + 1.80i·17-s + ⋯
L(s)  = 1  − 0.492·2-s + 0.577·3-s − 0.757·4-s − 0.591i·5-s − 0.284·6-s + 0.377i·7-s + 0.864·8-s + 0.333·9-s + 0.291i·10-s + 0.149i·11-s − 0.437·12-s − 1.58·13-s − 0.185i·14-s − 0.341i·15-s + 0.332·16-s + 0.106i·17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((−0.979−0.200i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1) L(s)=((−0.979−0.200i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: −0.979−0.200i-0.979 - 0.200i
Analytic conductor: 13.160713.1607
Root analytic conductor: 3.627783.62778
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ483(22,⋅)\chi_{483} (22, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1), −0.979−0.200i)(2,\ 483,\ (\ :1),\ -0.979 - 0.200i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0068027716240.006802771624
L(12)L(\frac12) ≈\approx 0.0068027716240.006802771624
L(2)L(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 1−1.73T 1 - 1.73T
7 1−2.64iT 1 - 2.64iT
23 1+(4.61−22.5i)T 1 + (4.61 - 22.5i)T
good2 1+0.984T+4T2 1 + 0.984T + 4T^{2}
5 1+2.95iT−25T2 1 + 2.95iT - 25T^{2}
11 1−1.64iT−121T2 1 - 1.64iT - 121T^{2}
13 1+20.6T+169T2 1 + 20.6T + 169T^{2}
17 1−1.80iT−289T2 1 - 1.80iT - 289T^{2}
19 1+30.5iT−361T2 1 + 30.5iT - 361T^{2}
29 1+23.0T+841T2 1 + 23.0T + 841T^{2}
31 1+46.3T+961T2 1 + 46.3T + 961T^{2}
37 1−61.7iT−1.36e3T2 1 - 61.7iT - 1.36e3T^{2}
41 1+46.0T+1.68e3T2 1 + 46.0T + 1.68e3T^{2}
43 1−47.1iT−1.84e3T2 1 - 47.1iT - 1.84e3T^{2}
47 1+77.1T+2.20e3T2 1 + 77.1T + 2.20e3T^{2}
53 1+75.2iT−2.80e3T2 1 + 75.2iT - 2.80e3T^{2}
59 1+59.3T+3.48e3T2 1 + 59.3T + 3.48e3T^{2}
61 1+13.6iT−3.72e3T2 1 + 13.6iT - 3.72e3T^{2}
67 1+24.4iT−4.48e3T2 1 + 24.4iT - 4.48e3T^{2}
71 1+24.4T+5.04e3T2 1 + 24.4T + 5.04e3T^{2}
73 1+5.03T+5.32e3T2 1 + 5.03T + 5.32e3T^{2}
79 1+120.iT−6.24e3T2 1 + 120. iT - 6.24e3T^{2}
83 1−84.7iT−6.88e3T2 1 - 84.7iT - 6.88e3T^{2}
89 1−62.8iT−7.92e3T2 1 - 62.8iT - 7.92e3T^{2}
97 1+49.7iT−9.40e3T2 1 + 49.7iT - 9.40e3T^{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

−11.08429527950794124134206916632, −9.755334812918075526876101537847, −9.473858624215313780108587392304, −8.632189965625352981490990496368, −7.80547473430492519376896119792, −6.92216162665573953992695156110, −5.13555752060625131960487022283, −4.71902673743281691485566320792, −3.21572502277770580469742616645, −1.72526143392981970156427337009, 0.00300831279112797365516213997, 1.91448250031726429109686698545, 3.38350823577132294086697434302, 4.39305022096838214647645009555, 5.55265568225939261945130921530, 7.09531371638641124322038303506, 7.65003780968628083199729702233, 8.611000945728733094382481918367, 9.505649019472820800870335974827, 10.19416725999346437328548031567

Graph of the ZZ-function along the critical line