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Properties

Label 2-1638-13.10-c1-0-14
Degree 22
Conductor 16381638
Sign 0.902−0.431i0.902 - 0.431i
Analytic cond. 13.079413.0794
Root an. cond. 3.616553.61655
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + 0.732i·5-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (0.366 + 0.633i)10-s + (−1.5 + 0.866i)11-s + (1.59 + 3.23i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (−1.86 + 3.23i)17-s + (0.866 + 0.5i)19-s + (0.633 + 0.366i)20-s + (−0.866 + 1.5i)22-s + (1.73 + 3i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.327i·5-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (0.115 + 0.200i)10-s + (−0.452 + 0.261i)11-s + (0.443 + 0.896i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.452 + 0.783i)17-s + (0.198 + 0.114i)19-s + (0.141 + 0.0818i)20-s + (−0.184 + 0.319i)22-s + (0.361 + 0.625i)23-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((0.902−0.431i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((0.902−0.431i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.902−0.431i0.902 - 0.431i
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1638(127,⋅)\chi_{1638} (127, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1638, ( :1/2), 0.902−0.431i)(2,\ 1638,\ (\ :1/2),\ 0.902 - 0.431i)

Particular Values

L(1)L(1) ≈\approx 2.4475767862.447576786
L(12)L(\frac12) ≈\approx 2.4475767862.447576786
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)
bad2 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
3 1 1
7 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
13 1+(−1.59−3.23i)T 1 + (-1.59 - 3.23i)T
good5 1−0.732iT−5T2 1 - 0.732iT - 5T^{2}
11 1+(1.5−0.866i)T+(5.5−9.52i)T2 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2}
17 1+(1.86−3.23i)T+(−8.5−14.7i)T2 1 + (1.86 - 3.23i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.866−0.5i)T+(9.5+16.4i)T2 1 + (-0.866 - 0.5i)T + (9.5 + 16.4i)T^{2}
23 1+(−1.73−3i)T+(−11.5+19.9i)T2 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.23−5.59i)T+(−14.5+25.1i)T2 1 + (-3.23 - 5.59i)T + (-14.5 + 25.1i)T^{2}
31 1+2.19iT−31T2 1 + 2.19iT - 31T^{2}
37 1+(−5.83+3.36i)T+(18.5−32.0i)T2 1 + (-5.83 + 3.36i)T + (18.5 - 32.0i)T^{2}
41 1+(2.59−1.5i)T+(20.5−35.5i)T2 1 + (2.59 - 1.5i)T + (20.5 - 35.5i)T^{2}
43 1+(−1.63+2.83i)T+(−21.5−37.2i)T2 1 + (-1.63 + 2.83i)T + (-21.5 - 37.2i)T^{2}
47 1−2.46iT−47T2 1 - 2.46iT - 47T^{2}
53 1+7T+53T2 1 + 7T + 53T^{2}
59 1+(0.803+0.464i)T+(29.5+51.0i)T2 1 + (0.803 + 0.464i)T + (29.5 + 51.0i)T^{2}
61 1+(2.59−4.5i)T+(−30.5−52.8i)T2 1 + (2.59 - 4.5i)T + (-30.5 - 52.8i)T^{2}
67 1+(−7.73+4.46i)T+(33.5−58.0i)T2 1 + (-7.73 + 4.46i)T + (33.5 - 58.0i)T^{2}
71 1+(−1.90−1.09i)T+(35.5+61.4i)T2 1 + (-1.90 - 1.09i)T + (35.5 + 61.4i)T^{2}
73 1−5.46iT−73T2 1 - 5.46iT - 73T^{2}
79 1−2.07T+79T2 1 - 2.07T + 79T^{2}
83 1−0.196iT−83T2 1 - 0.196iT - 83T^{2}
89 1+(−9.06+5.23i)T+(44.5−77.0i)T2 1 + (-9.06 + 5.23i)T + (44.5 - 77.0i)T^{2}
97 1+(−13.5−7.83i)T+(48.5+84.0i)T2 1 + (-13.5 - 7.83i)T + (48.5 + 84.0i)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

−9.462738210152191930777166342848, −8.748472342323068944538599892490, −7.77922849832951588386887083898, −6.86640724161462779695001214023, −6.17951509823235513325760044994, −5.20216383909316755572750853331, −4.42720917593724661239371599312, −3.48535681781101575635676774862, −2.45786563439892253109680955057, −1.42045467599086733677418857244, 0.824991909797718461465940940955, 2.50413742604742807317364036985, 3.35061895603064059981480144332, 4.59407574713558939662602346191, 5.05343742399378135050588325518, 6.04674905165356563452160751159, 6.81412061440473885144137882358, 7.79502036405850901701829937824, 8.324937270058030442492900953121, 9.167180307965718439152518910441

Graph of the ZZ-function along the critical line