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Properties

Label 2-1638-13.10-c1-0-17
Degree 22
Conductor 16381638
Sign 0.994−0.105i0.994 - 0.105i
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 + 0.901i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−0.450 − 0.781i)10-s + (3.75 − 2.16i)11-s + (−0.426 − 3.58i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (−2.53 + 4.38i)17-s + (5.34 + 3.08i)19-s + (0.781 + 0.450i)20-s + (−2.16 + 3.75i)22-s + (−4.22 − 7.31i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.403i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.142 − 0.246i)10-s + (1.13 − 0.653i)11-s + (−0.118 − 0.992i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.614 + 1.06i)17-s + (1.22 + 0.708i)19-s + (0.174 + 0.100i)20-s + (−0.462 + 0.800i)22-s + (−0.881 − 1.52i)23-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((0.994−0.105i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((0.994−0.105i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.994−0.105i0.994 - 0.105i
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), 0.994−0.105i)(2,\ 1638,\ (\ :1/2),\ 0.994 - 0.105i)

Particular Values

L(1)L(1) ≈\approx 1.3875513261.387551326
L(12)L(\frac12) ≈\approx 1.3875513261.387551326
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+(0.426+3.58i)T 1 + (0.426 + 3.58i)T
good5 1−0.901iT−5T2 1 - 0.901iT - 5T^{2}
11 1+(−3.75+2.16i)T+(5.5−9.52i)T2 1 + (-3.75 + 2.16i)T + (5.5 - 9.52i)T^{2}
17 1+(2.53−4.38i)T+(−8.5−14.7i)T2 1 + (2.53 - 4.38i)T + (-8.5 - 14.7i)T^{2}
19 1+(−5.34−3.08i)T+(9.5+16.4i)T2 1 + (-5.34 - 3.08i)T + (9.5 + 16.4i)T^{2}
23 1+(4.22+7.31i)T+(−11.5+19.9i)T2 1 + (4.22 + 7.31i)T + (-11.5 + 19.9i)T^{2}
29 1+(1.09+1.89i)T+(−14.5+25.1i)T2 1 + (1.09 + 1.89i)T + (-14.5 + 25.1i)T^{2}
31 1+0.873iT−31T2 1 + 0.873iT - 31T^{2}
37 1+(−0.124+0.0721i)T+(18.5−32.0i)T2 1 + (-0.124 + 0.0721i)T + (18.5 - 32.0i)T^{2}
41 1+(−3.46+1.99i)T+(20.5−35.5i)T2 1 + (-3.46 + 1.99i)T + (20.5 - 35.5i)T^{2}
43 1+(−3.85+6.67i)T+(−21.5−37.2i)T2 1 + (-3.85 + 6.67i)T + (-21.5 - 37.2i)T^{2}
47 1+2.92iT−47T2 1 + 2.92iT - 47T^{2}
53 1+1.69T+53T2 1 + 1.69T + 53T^{2}
59 1+(−7.40−4.27i)T+(29.5+51.0i)T2 1 + (-7.40 - 4.27i)T + (29.5 + 51.0i)T^{2}
61 1+(4.16−7.21i)T+(−30.5−52.8i)T2 1 + (4.16 - 7.21i)T + (-30.5 - 52.8i)T^{2}
67 1+(8.99−5.19i)T+(33.5−58.0i)T2 1 + (8.99 - 5.19i)T + (33.5 - 58.0i)T^{2}
71 1+(−2.83−1.63i)T+(35.5+61.4i)T2 1 + (-2.83 - 1.63i)T + (35.5 + 61.4i)T^{2}
73 1+0.539iT−73T2 1 + 0.539iT - 73T^{2}
79 1−6.53T+79T2 1 - 6.53T + 79T^{2}
83 1+13.2iT−83T2 1 + 13.2iT - 83T^{2}
89 1+(−6.74+3.89i)T+(44.5−77.0i)T2 1 + (-6.74 + 3.89i)T + (44.5 - 77.0i)T^{2}
97 1+(−10.1−5.85i)T+(48.5+84.0i)T2 1 + (-10.1 - 5.85i)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

−9.193650114727170519840168576555, −8.591441250887337449988616805023, −7.916731067416108357431290060827, −7.05829557173326216903647603756, −6.14701539415968648002954986691, −5.67877965827264367883147574496, −4.37120304835677143173640720982, −3.36485186679785775082858627651, −2.16079708030022018615363977058, −0.842534623739461893395076219275, 1.05992034725063246260820329733, 1.99903119263241178157146622856, 3.29634391080080342716618316358, 4.36811206288394672459834353553, 5.03735024868166052596681010909, 6.38265313561375339385787752714, 7.17478549824857992082557148038, 7.70932413740074769062901828798, 8.924328192734130654379720938683, 9.364754563557655780406274091985

Graph of the ZZ-function along the critical line