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Properties

Label 2-1440-160.19-c0-0-0
Degree 22
Conductor 14401440
Sign 0.555+0.831i0.555 + 0.831i
Analytic cond. 0.7186530.718653
Root an. cond. 0.8477340.847734
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  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (0.382 + 0.923i)5-s + (−0.923 + 0.382i)8-s + 10-s + i·16-s + 1.84·17-s + (0.707 − 1.70i)19-s + (0.382 − 0.923i)20-s + (0.541 + 0.541i)23-s + (−0.707 + 0.707i)25-s − 1.41i·31-s + (0.923 + 0.382i)32-s + (0.707 − 1.70i)34-s + (−1.30 − 1.30i)38-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (−0.707 − 0.707i)4-s + (0.382 + 0.923i)5-s + (−0.923 + 0.382i)8-s + 10-s + i·16-s + 1.84·17-s + (0.707 − 1.70i)19-s + (0.382 − 0.923i)20-s + (0.541 + 0.541i)23-s + (−0.707 + 0.707i)25-s − 1.41i·31-s + (0.923 + 0.382i)32-s + (0.707 − 1.70i)34-s + (−1.30 − 1.30i)38-s + ⋯

Functional equation

Λ(s)=(1440s/2 ΓC(s) L(s)=((0.555+0.831i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1440s/2 ΓC(s) L(s)=((0.555+0.831i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14401440    =    25⋅32⋅52^{5} \cdot 3^{2} \cdot 5
Sign: 0.555+0.831i0.555 + 0.831i
Analytic conductor: 0.7186530.718653
Root analytic conductor: 0.8477340.847734
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1440(19,⋅)\chi_{1440} (19, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1440, ( :0), 0.555+0.831i)(2,\ 1440,\ (\ :0),\ 0.555 + 0.831i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.3468377211.346837721
L(12)L(\frac12) ≈\approx 1.3468377211.346837721
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+(−0.382+0.923i)T 1 + (-0.382 + 0.923i)T
3 1 1
5 1+(−0.382−0.923i)T 1 + (-0.382 - 0.923i)T
good7 1−iT2 1 - iT^{2}
11 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
13 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
17 1−1.84T+T2 1 - 1.84T + T^{2}
19 1+(−0.707+1.70i)T+(−0.707−0.707i)T2 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2}
23 1+(−0.541−0.541i)T+iT2 1 + (-0.541 - 0.541i)T + iT^{2}
29 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
31 1+1.41iT−T2 1 + 1.41iT - T^{2}
37 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
41 1+iT2 1 + iT^{2}
43 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
47 1−1.84iT−T2 1 - 1.84iT - T^{2}
53 1+(1.30−0.541i)T+(0.707−0.707i)T2 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2}
59 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
61 1+(−0.292+0.707i)T+(−0.707−0.707i)T2 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2}
67 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
71 1+iT2 1 + iT^{2}
73 1−iT2 1 - iT^{2}
79 1+2T+T2 1 + 2T + T^{2}
83 1+(1.30+0.541i)T+(0.707+0.707i)T2 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2}
89 1−iT2 1 - iT^{2}
97 1−T2 1 - 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

−9.531555225639342006175746437668, −9.399904678785301262628204516463, −7.950480890242577609284118807644, −7.17070348572596042979740311329, −6.06336744428075710689591055225, −5.43551914263728054837150957472, −4.40616895923143626590892893629, −3.19223574811498754235292801754, −2.75411192494407367358931728483, −1.32239118898914464723366804000, 1.35326322453470757397759246155, 3.17184652890364860468629258192, 4.01966055549181989290746870474, 5.29051211760914600236031270383, 5.43445196039847682003895836933, 6.50378277082367548913192166125, 7.48493252972116377867043804991, 8.219626761502157695984613077385, 8.773609955791345383231418163240, 9.822545425116588180613369880299

Graph of the ZZ-function along the critical line