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Properties

Label 2-1638-13.10-c1-0-18
Degree 22
Conductor 16381638
Sign 0.967+0.252i0.967 + 0.252i
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 + 0.732i·5-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (0.366 + 0.633i)10-s + (3.23 − 1.86i)11-s + (−0.866 + 3.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (−0.133 + 0.232i)17-s + (3.86 + 2.23i)19-s + (0.633 + 0.366i)20-s + (1.86 − 3.23i)22-s + (−1.73 − 3i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.327i·5-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (0.115 + 0.200i)10-s + (0.974 − 0.562i)11-s + (−0.240 + 0.970i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.0324 + 0.0562i)17-s + (0.886 + 0.512i)19-s + (0.141 + 0.0818i)20-s + (0.397 − 0.689i)22-s + (−0.361 − 0.625i)23-s + ⋯

Functional equation

Λ(s)=(1638s/2 ΓC(s) L(s)=((0.967+0.252i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1638s/2 ΓC(s+1/2) L(s)=((0.967+0.252i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16381638    =    2⋅32⋅7⋅132 \cdot 3^{2} \cdot 7 \cdot 13
Sign: 0.967+0.252i0.967 + 0.252i
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), 0.967+0.252i)(2,\ 1638,\ (\ :1/2),\ 0.967 + 0.252i)

Particular Values

L(1)L(1) ≈\approx 2.7502587542.750258754
L(12)L(\frac12) ≈\approx 2.7502587542.750258754
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+(0.866−3.5i)T 1 + (0.866 - 3.5i)T
good5 1−0.732iT−5T2 1 - 0.732iT - 5T^{2}
11 1+(−3.23+1.86i)T+(5.5−9.52i)T2 1 + (-3.23 + 1.86i)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+(−3.86−2.23i)T+(9.5+16.4i)T2 1 + (-3.86 - 2.23i)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+(1.5+2.59i)T+(−14.5+25.1i)T2 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2}
31 1−7.66iT−31T2 1 - 7.66iT - 31T^{2}
37 1+(−1.09+0.633i)T+(18.5−32.0i)T2 1 + (-1.09 + 0.633i)T + (18.5 - 32.0i)T^{2}
41 1+(−6.06+3.5i)T+(20.5−35.5i)T2 1 + (-6.06 + 3.5i)T + (20.5 - 35.5i)T^{2}
43 1+(−0.366+0.633i)T+(−21.5−37.2i)T2 1 + (-0.366 + 0.633i)T + (-21.5 - 37.2i)T^{2}
47 1−4.46iT−47T2 1 - 4.46iT - 47T^{2}
53 1−10.4T+53T2 1 - 10.4T + 53T^{2}
59 1+(0.803+0.464i)T+(29.5+51.0i)T2 1 + (0.803 + 0.464i)T + (29.5 + 51.0i)T^{2}
61 1+(−5.86+10.1i)T+(−30.5−52.8i)T2 1 + (-5.86 + 10.1i)T + (-30.5 - 52.8i)T^{2}
67 1+(11.1−6.46i)T+(33.5−58.0i)T2 1 + (11.1 - 6.46i)T + (33.5 - 58.0i)T^{2}
71 1+(1.90+1.09i)T+(35.5+61.4i)T2 1 + (1.90 + 1.09i)T + (35.5 + 61.4i)T^{2}
73 1+6.53iT−73T2 1 + 6.53iT - 73T^{2}
79 1−10.8T+79T2 1 - 10.8T + 79T^{2}
83 1+5.66iT−83T2 1 + 5.66iT - 83T^{2}
89 1+(5.59−3.23i)T+(44.5−77.0i)T2 1 + (5.59 - 3.23i)T + (44.5 - 77.0i)T^{2}
97 1+(−0.633−0.366i)T+(48.5+84.0i)T2 1 + (-0.633 - 0.366i)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

−9.301775824541331496088061455174, −8.764943701750530599666094575509, −7.62812638722583085043280304320, −6.75576668788393758621570743412, −6.11184067962316252122264481034, −5.15154076030173980023054868720, −4.23679128141635827233325340439, −3.43688039242239385669896405475, −2.35630445313236814519010726077, −1.21094993674578404159716550300, 1.09635316575717931541341204383, 2.51548100720292289646161254086, 3.63783633997904440292916259931, 4.48742666992015819705964100624, 5.28515527781232310540471930832, 6.04240685948701470741419830826, 7.13794360000943123884101854617, 7.57119862756372242509886631496, 8.552191533719534780887513450056, 9.357055562636567038969532098530

Graph of the ZZ-function along the critical line