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Properties

Label 2-483-23.22-c2-0-35
Degree 22
Conductor 483483
Sign 0.980+0.195i0.980 + 0.195i
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  + 3.67·2-s − 1.73·3-s + 9.51·4-s − 1.72i·5-s − 6.36·6-s − 2.64i·7-s + 20.2·8-s + 2.99·9-s − 6.33i·10-s + 17.8i·11-s − 16.4·12-s + 23.2·13-s − 9.72i·14-s + 2.98i·15-s + 36.4·16-s − 19.8i·17-s + ⋯
L(s)  = 1  + 1.83·2-s − 0.577·3-s + 2.37·4-s − 0.344i·5-s − 1.06·6-s − 0.377i·7-s + 2.53·8-s + 0.333·9-s − 0.633i·10-s + 1.62i·11-s − 1.37·12-s + 1.78·13-s − 0.694i·14-s + 0.198i·15-s + 2.27·16-s − 1.16i·17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.980+0.195i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1) L(s)=((0.980+0.195i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.980+0.195i0.980 + 0.195i
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), 0.980+0.195i)(2,\ 483,\ (\ :1),\ 0.980 + 0.195i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 4.7501215334.750121533
L(12)L(\frac12) ≈\approx 4.7501215334.750121533
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+(−4.49+22.5i)T 1 + (-4.49 + 22.5i)T
good2 1−3.67T+4T2 1 - 3.67T + 4T^{2}
5 1+1.72iT−25T2 1 + 1.72iT - 25T^{2}
11 1−17.8iT−121T2 1 - 17.8iT - 121T^{2}
13 1−23.2T+169T2 1 - 23.2T + 169T^{2}
17 1+19.8iT−289T2 1 + 19.8iT - 289T^{2}
19 1+10.9iT−361T2 1 + 10.9iT - 361T^{2}
29 1+25.8T+841T2 1 + 25.8T + 841T^{2}
31 1+27.7T+961T2 1 + 27.7T + 961T^{2}
37 1−61.2iT−1.36e3T2 1 - 61.2iT - 1.36e3T^{2}
41 1−10.7T+1.68e3T2 1 - 10.7T + 1.68e3T^{2}
43 1+19.1iT−1.84e3T2 1 + 19.1iT - 1.84e3T^{2}
47 1+40.2T+2.20e3T2 1 + 40.2T + 2.20e3T^{2}
53 1−87.8iT−2.80e3T2 1 - 87.8iT - 2.80e3T^{2}
59 1+87.9T+3.48e3T2 1 + 87.9T + 3.48e3T^{2}
61 1−80.0iT−3.72e3T2 1 - 80.0iT - 3.72e3T^{2}
67 1−31.0iT−4.48e3T2 1 - 31.0iT - 4.48e3T^{2}
71 1+46.0T+5.04e3T2 1 + 46.0T + 5.04e3T^{2}
73 1+84.2T+5.32e3T2 1 + 84.2T + 5.32e3T^{2}
79 1+101.iT−6.24e3T2 1 + 101. iT - 6.24e3T^{2}
83 1−106.iT−6.88e3T2 1 - 106. iT - 6.88e3T^{2}
89 1+85.1iT−7.92e3T2 1 + 85.1iT - 7.92e3T^{2}
97 1+12.6iT−9.40e3T2 1 + 12.6iT - 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

−11.10571094712766904675436965739, −10.34286821742796285780458064211, −9.002458442875594894141346731464, −7.39814682326540242283577929516, −6.78355294929494204795937517702, −5.85241061498483635917735278017, −4.77491088504896760790266081959, −4.33972806839093286710403294947, −3.01726358393163580018145132079, −1.48986157988840549205358630514, 1.62846746555243034564852553727, 3.39862100347604174286351474996, 3.75710353938347465687704846806, 5.32521546313882426152255965719, 5.98637816054140135835679317724, 6.40270344002285700116915975399, 7.76688529606524437157753226497, 8.907869685792389335044248071317, 10.71049167433129668350487261777, 11.05105110463729454716786621273

Graph of the ZZ-function along the critical line