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Properties

Label 2-1638-13.10-c1-0-26
Degree 22
Conductor 16381638
Sign 0.923+0.383i0.923 + 0.383i
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 + 4.11i·5-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (−2.05 − 3.56i)10-s + (5.41 − 3.12i)11-s + (1.33 − 3.34i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (2.23 − 3.86i)17-s + (−2.94 − 1.70i)19-s + (3.56 + 2.05i)20-s + (−3.12 + 5.41i)22-s + (−1.49 − 2.58i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 1.84i·5-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (−0.651 − 1.12i)10-s + (1.63 − 0.942i)11-s + (0.371 − 0.928i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (0.541 − 0.938i)17-s + (−0.675 − 0.390i)19-s + (0.797 + 0.460i)20-s + (−0.666 + 1.15i)22-s + (−0.311 − 0.539i)23-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((0.923+0.383i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((0.923+0.383i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.923+0.383i0.923 + 0.383i
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), 0.923+0.383i)(2,\ 1638,\ (\ :1/2),\ 0.923 + 0.383i)

Particular Values

L(1)L(1) ≈\approx 1.1102679061.110267906
L(12)L(\frac12) ≈\approx 1.1102679061.110267906
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+(−1.33+3.34i)T 1 + (-1.33 + 3.34i)T
good5 1−4.11iT−5T2 1 - 4.11iT - 5T^{2}
11 1+(−5.41+3.12i)T+(5.5−9.52i)T2 1 + (-5.41 + 3.12i)T + (5.5 - 9.52i)T^{2}
17 1+(−2.23+3.86i)T+(−8.5−14.7i)T2 1 + (-2.23 + 3.86i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.94+1.70i)T+(9.5+16.4i)T2 1 + (2.94 + 1.70i)T + (9.5 + 16.4i)T^{2}
23 1+(1.49+2.58i)T+(−11.5+19.9i)T2 1 + (1.49 + 2.58i)T + (-11.5 + 19.9i)T^{2}
29 1+(2.51+4.35i)T+(−14.5+25.1i)T2 1 + (2.51 + 4.35i)T + (-14.5 + 25.1i)T^{2}
31 1+10.7iT−31T2 1 + 10.7iT - 31T^{2}
37 1+(−3.52+2.03i)T+(18.5−32.0i)T2 1 + (-3.52 + 2.03i)T + (18.5 - 32.0i)T^{2}
41 1+(5.96−3.44i)T+(20.5−35.5i)T2 1 + (5.96 - 3.44i)T + (20.5 - 35.5i)T^{2}
43 1+(1.83−3.17i)T+(−21.5−37.2i)T2 1 + (1.83 - 3.17i)T + (-21.5 - 37.2i)T^{2}
47 1+9.29iT−47T2 1 + 9.29iT - 47T^{2}
53 1−1.32T+53T2 1 - 1.32T + 53T^{2}
59 1+(−2.81−1.62i)T+(29.5+51.0i)T2 1 + (-2.81 - 1.62i)T + (29.5 + 51.0i)T^{2}
61 1+(0.550−0.953i)T+(−30.5−52.8i)T2 1 + (0.550 - 0.953i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.360+0.208i)T+(33.5−58.0i)T2 1 + (-0.360 + 0.208i)T + (33.5 - 58.0i)T^{2}
71 1+(−13.1−7.61i)T+(35.5+61.4i)T2 1 + (-13.1 - 7.61i)T + (35.5 + 61.4i)T^{2}
73 1+11.7iT−73T2 1 + 11.7iT - 73T^{2}
79 1−1.31T+79T2 1 - 1.31T + 79T^{2}
83 1+8.04iT−83T2 1 + 8.04iT - 83T^{2}
89 1+(8.60−4.96i)T+(44.5−77.0i)T2 1 + (8.60 - 4.96i)T + (44.5 - 77.0i)T^{2}
97 1+(10.5+6.07i)T+(48.5+84.0i)T2 1 + (10.5 + 6.07i)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

−9.532709239073225586928136273249, −8.432761849569828953929751523503, −7.67833455598890971352795063347, −6.85377869684081747328917692489, −6.31248416180033278478115554253, −5.75353791518343012983356947254, −4.01269597838947612820567713090, −3.26399926745146487215483505997, −2.30870770245819443981073822885, −0.56128592125252866826662667249, 1.39958233187838781177203090678, 1.68243821042084537098837174117, 3.70141404359133002302889539655, 4.22903852715136845753321174626, 5.24399175902788642328552978285, 6.31278656965280178936248882938, 7.06760343098430974736637467441, 8.319607208153126650979382619925, 8.673360464868108279538611960780, 9.432977419635513497496273459163

Graph of the ZZ-function along the critical line