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Properties

Label 2-483-7.2-c1-0-8
Degree 22
Conductor 483483
Sign 0.914−0.404i0.914 - 0.404i
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  + (1.13 − 1.96i)2-s + (0.5 + 0.866i)3-s + (−1.58 − 2.74i)4-s + (−2.07 + 3.59i)5-s + 2.27·6-s + (−1.69 + 2.03i)7-s − 2.64·8-s + (−0.499 + 0.866i)9-s + (4.71 + 8.17i)10-s + (1.59 + 2.76i)11-s + (1.58 − 2.74i)12-s + 0.175·13-s + (2.08 + 5.64i)14-s − 4.15·15-s + (0.156 − 0.270i)16-s + (0.663 + 1.14i)17-s + ⋯
L(s)  = 1  + (0.803 − 1.39i)2-s + (0.288 + 0.499i)3-s + (−0.791 − 1.37i)4-s + (−0.928 + 1.60i)5-s + 0.927·6-s + (−0.639 + 0.768i)7-s − 0.936·8-s + (−0.166 + 0.288i)9-s + (1.49 + 2.58i)10-s + (0.480 + 0.832i)11-s + (0.456 − 0.791i)12-s + 0.0487·13-s + (0.556 + 1.50i)14-s − 1.07·15-s + (0.0390 − 0.0676i)16-s + (0.160 + 0.278i)17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.914−0.404i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.914−0.404i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.914−0.404i0.914 - 0.404i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(415,⋅)\chi_{483} (415, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.914−0.404i)(2,\ 483,\ (\ :1/2),\ 0.914 - 0.404i)

Particular Values

L(1)L(1) ≈\approx 1.67621+0.354493i1.67621 + 0.354493i
L(12)L(\frac12) ≈\approx 1.67621+0.354493i1.67621 + 0.354493i
L(32)L(\frac{3}{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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1+(1.69−2.03i)T 1 + (1.69 - 2.03i)T
23 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good2 1+(−1.13+1.96i)T+(−1−1.73i)T2 1 + (-1.13 + 1.96i)T + (-1 - 1.73i)T^{2}
5 1+(2.07−3.59i)T+(−2.5−4.33i)T2 1 + (2.07 - 3.59i)T + (-2.5 - 4.33i)T^{2}
11 1+(−1.59−2.76i)T+(−5.5+9.52i)T2 1 + (-1.59 - 2.76i)T + (-5.5 + 9.52i)T^{2}
13 1−0.175T+13T2 1 - 0.175T + 13T^{2}
17 1+(−0.663−1.14i)T+(−8.5+14.7i)T2 1 + (-0.663 - 1.14i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.0252−0.0436i)T+(−9.5−16.4i)T2 1 + (0.0252 - 0.0436i)T + (-9.5 - 16.4i)T^{2}
29 1−1.10T+29T2 1 - 1.10T + 29T^{2}
31 1+(−4.35−7.54i)T+(−15.5+26.8i)T2 1 + (-4.35 - 7.54i)T + (-15.5 + 26.8i)T^{2}
37 1+(1.88−3.25i)T+(−18.5−32.0i)T2 1 + (1.88 - 3.25i)T + (-18.5 - 32.0i)T^{2}
41 1−4.84T+41T2 1 - 4.84T + 41T^{2}
43 1+10.5T+43T2 1 + 10.5T + 43T^{2}
47 1+(−4.47+7.75i)T+(−23.5−40.7i)T2 1 + (-4.47 + 7.75i)T + (-23.5 - 40.7i)T^{2}
53 1+(−4.98−8.63i)T+(−26.5+45.8i)T2 1 + (-4.98 - 8.63i)T + (-26.5 + 45.8i)T^{2}
59 1+(−3.19−5.53i)T+(−29.5+51.0i)T2 1 + (-3.19 - 5.53i)T + (-29.5 + 51.0i)T^{2}
61 1+(−3.61+6.26i)T+(−30.5−52.8i)T2 1 + (-3.61 + 6.26i)T + (-30.5 - 52.8i)T^{2}
67 1+(3.01+5.21i)T+(−33.5+58.0i)T2 1 + (3.01 + 5.21i)T + (-33.5 + 58.0i)T^{2}
71 1−6.71T+71T2 1 - 6.71T + 71T^{2}
73 1+(−2.30−3.99i)T+(−36.5+63.2i)T2 1 + (-2.30 - 3.99i)T + (-36.5 + 63.2i)T^{2}
79 1+(−7.74+13.4i)T+(−39.5−68.4i)T2 1 + (-7.74 + 13.4i)T + (-39.5 - 68.4i)T^{2}
83 1−7.21T+83T2 1 - 7.21T + 83T^{2}
89 1+(−0.970+1.68i)T+(−44.5−77.0i)T2 1 + (-0.970 + 1.68i)T + (-44.5 - 77.0i)T^{2}
97 1+12.7T+97T2 1 + 12.7T + 97T^{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

−11.09630965071401166465835123607, −10.32231176298318989741328681899, −9.840737815364508812920517492763, −8.586262837046732801763725509156, −7.25862011868388112799848448388, −6.31288381194569976460868929205, −4.86361504532602463103856251409, −3.77459631805805611650195193353, −3.16416138126449895106517805066, −2.27501025536518949375066765892, 0.814090694750727083414636215371, 3.58743315325005279377968649464, 4.25695546895641842709474458823, 5.31697640186767448771558463839, 6.30683705125668241752220823999, 7.26819887392800533794004781951, 8.028603684231789253958443974596, 8.627259998433891908405580413631, 9.609712346021600688375234829901, 11.29123271838147190855394576787

Graph of the ZZ-function along the critical line