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Properties

Label 2-1638-13.10-c1-0-20
Degree 22
Conductor 16381638
Sign 0.0515+0.998i0.0515 + 0.998i
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 + 2.73i·5-s + (−0.866 − 0.5i)7-s + 0.999i·8-s + (−1.36 − 2.36i)10-s + (−1.5 + 0.866i)11-s + (−3.59 + 0.232i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (−0.133 + 0.232i)17-s + (−0.866 − 0.5i)19-s + (2.36 + 1.36i)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 + 1.22i·5-s + (−0.327 − 0.188i)7-s + 0.353i·8-s + (−0.431 − 0.748i)10-s + (−0.452 + 0.261i)11-s + (−0.997 + 0.0643i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.0324 + 0.0562i)17-s + (−0.198 − 0.114i)19-s + (0.529 + 0.305i)20-s + (0.184 − 0.319i)22-s + (−0.361 − 0.625i)23-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((0.0515+0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((0.0515+0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.0515+0.998i0.0515 + 0.998i
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), 0.0515+0.998i)(2,\ 1638,\ (\ :1/2),\ 0.0515 + 0.998i)

Particular Values

L(1)L(1) ≈\approx 0.30379123220.3037912322
L(12)L(\frac12) ≈\approx 0.30379123220.3037912322
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+(3.59−0.232i)T 1 + (3.59 - 0.232i)T
good5 1−2.73iT−5T2 1 - 2.73iT - 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+(0.133−0.232i)T+(−8.5−14.7i)T2 1 + (0.133 - 0.232i)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+(0.232+0.401i)T+(−14.5+25.1i)T2 1 + (0.232 + 0.401i)T + (-14.5 + 25.1i)T^{2}
31 1+8.19iT−31T2 1 + 8.19iT - 31T^{2}
37 1+(2.83−1.63i)T+(18.5−32.0i)T2 1 + (2.83 - 1.63i)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+(−3.36+5.83i)T+(−21.5−37.2i)T2 1 + (-3.36 + 5.83i)T + (-21.5 - 37.2i)T^{2}
47 1−4.46iT−47T2 1 - 4.46iT - 47T^{2}
53 1+7T+53T2 1 + 7T + 53T^{2}
59 1+(11.1+6.46i)T+(29.5+51.0i)T2 1 + (11.1 + 6.46i)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+(−4.26+2.46i)T+(33.5−58.0i)T2 1 + (-4.26 + 2.46i)T + (33.5 - 58.0i)T^{2}
71 1+(−7.09−4.09i)T+(35.5+61.4i)T2 1 + (-7.09 - 4.09i)T + (35.5 + 61.4i)T^{2}
73 1−1.46iT−73T2 1 - 1.46iT - 73T^{2}
79 1−15.9T+79T2 1 - 15.9T + 79T^{2}
83 1−10.1iT−83T2 1 - 10.1iT - 83T^{2}
89 1+(3.06−1.76i)T+(44.5−77.0i)T2 1 + (3.06 - 1.76i)T + (44.5 - 77.0i)T^{2}
97 1+(−1.43−0.830i)T+(48.5+84.0i)T2 1 + (-1.43 - 0.830i)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

−9.456446121383409796323187591175, −8.184006057328682855685433038098, −7.57776402313324679074127013781, −6.84061922299957008908848472846, −6.28088336786786603251600828170, −5.23590409029145978157722946742, −4.11825568629710228907554964164, −2.87819728676688643803261904369, −2.13941053305896401936935349682, −0.14828689520451778951719055200, 1.20931236325008293056005258602, 2.41310867188804024984334067339, 3.49525555904957893082338120066, 4.69257207666598838490961802242, 5.34070015264298279430265884688, 6.40829989080868768616117329519, 7.47281806995968213634845871003, 8.111190889429782803406423074805, 8.948583028908898092779748173376, 9.407970166554432703057398956086

Graph of the ZZ-function along the critical line