Properties

Label 2-483-21.20-c1-0-31
Degree 22
Conductor 483483
Sign 0.8720.487i0.872 - 0.487i
Analytic cond. 3.856773.85677
Root an. cond. 1.963861.96386
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.61i·2-s + (−0.618 − 1.61i)3-s − 0.618·4-s + 2.61·5-s + (2.61 − 1.00i)6-s + (2.61 − 0.381i)7-s + 2.23i·8-s + (−2.23 + 2.00i)9-s + 4.23i·10-s − 2.23i·11-s + (0.381 + 1.00i)12-s − 6.85i·13-s + (0.618 + 4.23i)14-s + (−1.61 − 4.23i)15-s − 4.85·16-s + 1.47·17-s + ⋯
L(s)  = 1  + 1.14i·2-s + (−0.356 − 0.934i)3-s − 0.309·4-s + 1.17·5-s + (1.06 − 0.408i)6-s + (0.989 − 0.144i)7-s + 0.790i·8-s + (−0.745 + 0.666i)9-s + 1.33i·10-s − 0.674i·11-s + (0.110 + 0.288i)12-s − 1.90i·13-s + (0.165 + 1.13i)14-s + (−0.417 − 1.09i)15-s − 1.21·16-s + 0.357·17-s + ⋯

Functional equation

Λ(s)=(483s/2ΓC(s)L(s)=((0.8720.487i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2ΓC(s+1/2)L(s)=((0.8720.487i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.487i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    37233 \cdot 7 \cdot 23
Sign: 0.8720.487i0.872 - 0.487i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(461,)\chi_{483} (461, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.8720.487i)(2,\ 483,\ (\ :1/2),\ 0.872 - 0.487i)

Particular Values

L(1)L(1) \approx 1.70537+0.444310i1.70537 + 0.444310i
L(12)L(\frac12) \approx 1.70537+0.444310i1.70537 + 0.444310i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(0.618+1.61i)T 1 + (0.618 + 1.61i)T
7 1+(2.61+0.381i)T 1 + (-2.61 + 0.381i)T
23 1+iT 1 + iT
good2 11.61iT2T2 1 - 1.61iT - 2T^{2}
5 12.61T+5T2 1 - 2.61T + 5T^{2}
11 1+2.23iT11T2 1 + 2.23iT - 11T^{2}
13 1+6.85iT13T2 1 + 6.85iT - 13T^{2}
17 11.47T+17T2 1 - 1.47T + 17T^{2}
19 15.47iT19T2 1 - 5.47iT - 19T^{2}
29 11.76iT29T2 1 - 1.76iT - 29T^{2}
31 10.527iT31T2 1 - 0.527iT - 31T^{2}
37 15.76T+37T2 1 - 5.76T + 37T^{2}
41 10.236T+41T2 1 - 0.236T + 41T^{2}
43 17.61T+43T2 1 - 7.61T + 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 113.5iT53T2 1 - 13.5iT - 53T^{2}
59 1+7.56T+59T2 1 + 7.56T + 59T^{2}
61 10.854iT61T2 1 - 0.854iT - 61T^{2}
67 1+10.6T+67T2 1 + 10.6T + 67T^{2}
71 1+5.32iT71T2 1 + 5.32iT - 71T^{2}
73 1+8.23iT73T2 1 + 8.23iT - 73T^{2}
79 1+7.76T+79T2 1 + 7.76T + 79T^{2}
83 1+10.7T+83T2 1 + 10.7T + 83T^{2}
89 18.09T+89T2 1 - 8.09T + 89T^{2}
97 1+1.94iT97T2 1 + 1.94iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.93903319445090684067843995684, −10.35366135591398325780194589712, −8.832238057650210038898710223558, −7.86770920276614059470928369557, −7.62623249791977015386424505931, −6.06400760783670640779205496331, −5.88950177272823048780742314086, −5.04290162688712199285765151897, −2.74318620388158491255135281557, −1.39647026846428147050162044608, 1.63055850491832581965360534497, 2.57099904858599971441612436658, 4.17227825748126203774238747866, 4.84315322848213187018152379581, 6.07842113778678663209393091182, 7.07449734467936911617856714257, 8.813734204348968379490012841369, 9.572211962557621922894317229963, 9.952290143113457973230213222877, 11.17991332144646222900145771611

Graph of the ZZ-function along the critical line