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Properties

Label 2-1638-13.4-c1-0-11
Degree 22
Conductor 16381638
Sign 0.419−0.907i0.419 - 0.907i
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 + 3.05i·5-s + (0.866 − 0.5i)7-s − 0.999i·8-s + (1.52 − 2.64i)10-s + (2.98 + 1.72i)11-s + (3.25 + 1.55i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (1.41 + 2.44i)17-s + (1.49 − 0.862i)19-s + (−2.64 + 1.52i)20-s + (−1.72 − 2.98i)22-s + (−1.53 + 2.65i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 1.36i·5-s + (0.327 − 0.188i)7-s − 0.353i·8-s + (0.483 − 0.836i)10-s + (0.898 + 0.518i)11-s + (0.902 + 0.431i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (0.343 + 0.594i)17-s + (0.342 − 0.197i)19-s + (−0.591 + 0.341i)20-s + (−0.366 − 0.635i)22-s + (−0.319 + 0.553i)23-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((0.419−0.907i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((0.419−0.907i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.419−0.907i0.419 - 0.907i
Analytic conductor: 13.079413.0794
Root analytic conductor: 3.616553.61655
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1638(1135,⋅)\chi_{1638} (1135, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1638, ( :1/2), 0.419−0.907i)(2,\ 1638,\ (\ :1/2),\ 0.419 - 0.907i)

Particular Values

L(1)L(1) ≈\approx 1.4258458361.425845836
L(12)L(\frac12) ≈\approx 1.4258458361.425845836
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+(−3.25−1.55i)T 1 + (-3.25 - 1.55i)T
good5 1−3.05iT−5T2 1 - 3.05iT - 5T^{2}
11 1+(−2.98−1.72i)T+(5.5+9.52i)T2 1 + (-2.98 - 1.72i)T + (5.5 + 9.52i)T^{2}
17 1+(−1.41−2.44i)T+(−8.5+14.7i)T2 1 + (-1.41 - 2.44i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1.49+0.862i)T+(9.5−16.4i)T2 1 + (-1.49 + 0.862i)T + (9.5 - 16.4i)T^{2}
23 1+(1.53−2.65i)T+(−11.5−19.9i)T2 1 + (1.53 - 2.65i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.92+3.33i)T+(−14.5−25.1i)T2 1 + (-1.92 + 3.33i)T + (-14.5 - 25.1i)T^{2}
31 1+0.978iT−31T2 1 + 0.978iT - 31T^{2}
37 1+(−4.58−2.64i)T+(18.5+32.0i)T2 1 + (-4.58 - 2.64i)T + (18.5 + 32.0i)T^{2}
41 1+(8.62+4.98i)T+(20.5+35.5i)T2 1 + (8.62 + 4.98i)T + (20.5 + 35.5i)T^{2}
43 1+(1.51+2.61i)T+(−21.5+37.2i)T2 1 + (1.51 + 2.61i)T + (-21.5 + 37.2i)T^{2}
47 1+8.04iT−47T2 1 + 8.04iT - 47T^{2}
53 1−8.33T+53T2 1 - 8.33T + 53T^{2}
59 1+(8.64−4.98i)T+(29.5−51.0i)T2 1 + (8.64 - 4.98i)T + (29.5 - 51.0i)T^{2}
61 1+(−5.77−9.99i)T+(−30.5+52.8i)T2 1 + (-5.77 - 9.99i)T + (-30.5 + 52.8i)T^{2}
67 1+(−4.11−2.37i)T+(33.5+58.0i)T2 1 + (-4.11 - 2.37i)T + (33.5 + 58.0i)T^{2}
71 1+(−3.17+1.83i)T+(35.5−61.4i)T2 1 + (-3.17 + 1.83i)T + (35.5 - 61.4i)T^{2}
73 1−10.1iT−73T2 1 - 10.1iT - 73T^{2}
79 1+4.49T+79T2 1 + 4.49T + 79T^{2}
83 1+7.15iT−83T2 1 + 7.15iT - 83T^{2}
89 1+(−6.84−3.95i)T+(44.5+77.0i)T2 1 + (-6.84 - 3.95i)T + (44.5 + 77.0i)T^{2}
97 1+(7.88−4.54i)T+(48.5−84.0i)T2 1 + (7.88 - 4.54i)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

−9.683748998090173868128419899773, −8.760408771434545661768903707342, −7.988765963411748305367563121158, −7.04081775513868856248831005029, −6.65073504214617368971132776691, −5.65450961735001072895215586486, −4.11920407765746561208177629165, −3.54134925062431481919316179599, −2.38585235550915691570208566319, −1.34799780408921185919779561263, 0.78667043121624930673587442428, 1.56985367691734936039870427237, 3.21325326088267648399852447054, 4.40473692229841694767368298466, 5.23708621441581528634199363297, 5.98871710116246763032637161638, 6.83460779957128155255915626031, 8.108657602204317950577141389559, 8.319904115835358225297619111674, 9.169371347876956374465194817525

Graph of the ZZ-function along the critical line