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Properties

Label 2-40e2-5.3-c0-0-1
Degree 22
Conductor 16001600
Sign 0.850−0.525i0.850 - 0.525i
Analytic cond. 0.7985040.798504
Root an. cond. 0.8935900.893590
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)3-s + (1 − i)7-s + i·9-s + 2·21-s + (−1 − i)23-s + 2i·29-s + (−1 − i)43-s + (−1 + i)47-s − i·49-s + (1 + i)63-s + (−1 + i)67-s − 2i·69-s + 81-s + (−1 − i)83-s + (−2 + 2i)87-s + ⋯
L(s)  = 1  + (1 + i)3-s + (1 − i)7-s + i·9-s + 2·21-s + (−1 − i)23-s + 2i·29-s + (−1 − i)43-s + (−1 + i)47-s − i·49-s + (1 + i)63-s + (−1 + i)67-s − 2i·69-s + 81-s + (−1 − i)83-s + (−2 + 2i)87-s + ⋯

Functional equation

Λ(s)=(1600s/2 ΓC(s) L(s)=((0.850−0.525i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1600s/2 ΓC(s) L(s)=((0.850−0.525i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 16001600    =    26⋅522^{6} \cdot 5^{2}
Sign: 0.850−0.525i0.850 - 0.525i
Analytic conductor: 0.7985040.798504
Root analytic conductor: 0.8935900.893590
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1600(193,⋅)\chi_{1600} (193, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1600, ( :0), 0.850−0.525i)(2,\ 1600,\ (\ :0),\ 0.850 - 0.525i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.6735535641.673553564
L(12)L(\frac12) ≈\approx 1.6735535641.673553564
L(1)L(1) 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 1
5 1 1
good3 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
7 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
11 1+T2 1 + T^{2}
13 1+iT2 1 + iT^{2}
17 1−iT2 1 - iT^{2}
19 1−T2 1 - T^{2}
23 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
29 1−2iT−T2 1 - 2iT - T^{2}
31 1+T2 1 + T^{2}
37 1−iT2 1 - iT^{2}
41 1+T2 1 + T^{2}
43 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
47 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
53 1+iT2 1 + iT^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
71 1+T2 1 + T^{2}
73 1+iT2 1 + iT^{2}
79 1−T2 1 - T^{2}
83 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
89 1−2iT−T2 1 - 2iT - T^{2}
97 1−iT2 1 - iT^{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

−9.702788932087076384253681683392, −8.768539091866812237486005072799, −8.290612172734234462516375814948, −7.49852891830371670949769420491, −6.59533025265282169763422979388, −5.22812367134743977701325713085, −4.48116622592691338547202548855, −3.84278008463460074942882744315, −2.89458068973806471730627222799, −1.59584208834128298357390072734, 1.64300569239688915817781434294, 2.24492217076563886293446112366, 3.27385646867688932235280580080, 4.50916375256853110461324208476, 5.55617438434589328247878086889, 6.35910984698418105891142023115, 7.42271215906803915216995867190, 8.081474969791895606429605224165, 8.424259432951252227075141732133, 9.336137767135761875116565430071

Graph of the ZZ-function along the critical line