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Properties

Label 2-1638-13.10-c1-0-22
Degree 22
Conductor 16381638
Sign 0.311+0.950i0.311 + 0.950i
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 + 0.332i·5-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.166 + 0.288i)10-s + (−2.26 + 1.30i)11-s + (3.41 − 1.16i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (1.94 − 3.36i)17-s + (4.85 + 2.80i)19-s + (0.288 + 0.166i)20-s + (−1.30 + 2.26i)22-s + (−2.10 − 3.64i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.148i·5-s + (−0.327 − 0.188i)7-s − 0.353i·8-s + (0.0526 + 0.0911i)10-s + (−0.681 + 0.393i)11-s + (0.946 − 0.323i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.471 − 0.816i)17-s + (1.11 + 0.643i)19-s + (0.0644 + 0.0372i)20-s + (−0.278 + 0.481i)22-s + (−0.439 − 0.760i)23-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((0.311+0.950i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((0.311+0.950i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.311+0.950i0.311 + 0.950i
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), 0.311+0.950i)(2,\ 1638,\ (\ :1/2),\ 0.311 + 0.950i)

Particular Values

L(1)L(1) ≈\approx 2.3510534282.351053428
L(12)L(\frac12) ≈\approx 2.3510534282.351053428
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+(−3.41+1.16i)T 1 + (-3.41 + 1.16i)T
good5 1−0.332iT−5T2 1 - 0.332iT - 5T^{2}
11 1+(2.26−1.30i)T+(5.5−9.52i)T2 1 + (2.26 - 1.30i)T + (5.5 - 9.52i)T^{2}
17 1+(−1.94+3.36i)T+(−8.5−14.7i)T2 1 + (-1.94 + 3.36i)T + (-8.5 - 14.7i)T^{2}
19 1+(−4.85−2.80i)T+(9.5+16.4i)T2 1 + (-4.85 - 2.80i)T + (9.5 + 16.4i)T^{2}
23 1+(2.10+3.64i)T+(−11.5+19.9i)T2 1 + (2.10 + 3.64i)T + (-11.5 + 19.9i)T^{2}
29 1+(0.593+1.02i)T+(−14.5+25.1i)T2 1 + (0.593 + 1.02i)T + (-14.5 + 25.1i)T^{2}
31 1+7.07iT−31T2 1 + 7.07iT - 31T^{2}
37 1+(−0.499+0.288i)T+(18.5−32.0i)T2 1 + (-0.499 + 0.288i)T + (18.5 - 32.0i)T^{2}
41 1+(0.451−0.260i)T+(20.5−35.5i)T2 1 + (0.451 - 0.260i)T + (20.5 - 35.5i)T^{2}
43 1+(−1.53+2.66i)T+(−21.5−37.2i)T2 1 + (-1.53 + 2.66i)T + (-21.5 - 37.2i)T^{2}
47 1+12.0iT−47T2 1 + 12.0iT - 47T^{2}
53 1−9.71T+53T2 1 - 9.71T + 53T^{2}
59 1+(9.07+5.23i)T+(29.5+51.0i)T2 1 + (9.07 + 5.23i)T + (29.5 + 51.0i)T^{2}
61 1+(3.71−6.44i)T+(−30.5−52.8i)T2 1 + (3.71 - 6.44i)T + (-30.5 - 52.8i)T^{2}
67 1+(−10.3+5.96i)T+(33.5−58.0i)T2 1 + (-10.3 + 5.96i)T + (33.5 - 58.0i)T^{2}
71 1+(−0.818−0.472i)T+(35.5+61.4i)T2 1 + (-0.818 - 0.472i)T + (35.5 + 61.4i)T^{2}
73 1−4.66iT−73T2 1 - 4.66iT - 73T^{2}
79 1+0.943T+79T2 1 + 0.943T + 79T^{2}
83 1−13.7iT−83T2 1 - 13.7iT - 83T^{2}
89 1+(2.92−1.69i)T+(44.5−77.0i)T2 1 + (2.92 - 1.69i)T + (44.5 - 77.0i)T^{2}
97 1+(5.03+2.90i)T+(48.5+84.0i)T2 1 + (5.03 + 2.90i)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

−9.467122341450245322681825679221, −8.367750895887518300530450217858, −7.54038928009900666139134357339, −6.74940014561860881513918390024, −5.77966528849822350646334928410, −5.15710476354905533571861542277, −4.04969479874292474326173540571, −3.23990902169170392037671910448, −2.30616552449167365414273615638, −0.819170151948762855511968623689, 1.33471838791788171415647508572, 2.89452496836120799673419166325, 3.54162698824933471898446459937, 4.66836609041800098213331958733, 5.53358998092128056145189061409, 6.14765154880555819688879936364, 7.07322499525216528596329389030, 7.88224074426394905918903894761, 8.668825064834070714053365438228, 9.377313244684780833146798927034

Graph of the ZZ-function along the critical line