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Properties

Label 2-960-960.749-c0-0-0
Degree 22
Conductor 960960
Sign 0.956−0.290i0.956 - 0.290i
Analytic cond. 0.4791020.479102
Root an. cond. 0.6921720.692172
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  + (−0.980 − 0.195i)2-s + (0.555 + 0.831i)3-s + (0.923 + 0.382i)4-s + (−0.195 − 0.980i)5-s + (−0.382 − 0.923i)6-s + (−0.831 − 0.555i)8-s + (−0.382 + 0.923i)9-s + i·10-s + (0.195 + 0.980i)12-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)16-s + (0.275 + 0.275i)17-s + (0.555 − 0.831i)18-s + (1.63 + 0.324i)19-s + (0.195 − 0.980i)20-s + ⋯
L(s)  = 1  + (−0.980 − 0.195i)2-s + (0.555 + 0.831i)3-s + (0.923 + 0.382i)4-s + (−0.195 − 0.980i)5-s + (−0.382 − 0.923i)6-s + (−0.831 − 0.555i)8-s + (−0.382 + 0.923i)9-s + i·10-s + (0.195 + 0.980i)12-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)16-s + (0.275 + 0.275i)17-s + (0.555 − 0.831i)18-s + (1.63 + 0.324i)19-s + (0.195 − 0.980i)20-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.956−0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s) L(s)=((0.956−0.290i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.956−0.290i0.956 - 0.290i
Analytic conductor: 0.4791020.479102
Root analytic conductor: 0.6921720.692172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ960(749,⋅)\chi_{960} (749, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :0), 0.956−0.290i)(2,\ 960,\ (\ :0),\ 0.956 - 0.290i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.79786379970.7978637997
L(12)L(\frac12) ≈\approx 0.79786379970.7978637997
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+(0.980+0.195i)T 1 + (0.980 + 0.195i)T
3 1+(−0.555−0.831i)T 1 + (-0.555 - 0.831i)T
5 1+(0.195+0.980i)T 1 + (0.195 + 0.980i)T
good7 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
11 1+(−0.382−0.923i)T2 1 + (-0.382 - 0.923i)T^{2}
13 1+(0.923+0.382i)T2 1 + (0.923 + 0.382i)T^{2}
17 1+(−0.275−0.275i)T+iT2 1 + (-0.275 - 0.275i)T + iT^{2}
19 1+(−1.63−0.324i)T+(0.923+0.382i)T2 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2}
23 1+(−1.81−0.750i)T+(0.707+0.707i)T2 1 + (-1.81 - 0.750i)T + (0.707 + 0.707i)T^{2}
29 1+(−0.382+0.923i)T2 1 + (-0.382 + 0.923i)T^{2}
31 1+1.84iT−T2 1 + 1.84iT - T^{2}
37 1+(−0.923+0.382i)T2 1 + (-0.923 + 0.382i)T^{2}
41 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
43 1+(0.382+0.923i)T2 1 + (0.382 + 0.923i)T^{2}
47 1+(−0.785−0.785i)T+iT2 1 + (-0.785 - 0.785i)T + iT^{2}
53 1+(0.636+0.425i)T+(0.382+0.923i)T2 1 + (0.636 + 0.425i)T + (0.382 + 0.923i)T^{2}
59 1+(0.923−0.382i)T2 1 + (0.923 - 0.382i)T^{2}
61 1+(0.923+1.38i)T+(−0.382+0.923i)T2 1 + (0.923 + 1.38i)T + (-0.382 + 0.923i)T^{2}
67 1+(0.382−0.923i)T2 1 + (0.382 - 0.923i)T^{2}
71 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
73 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
79 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
83 1+(0.750+0.149i)T+(0.923+0.382i)T2 1 + (0.750 + 0.149i)T + (0.923 + 0.382i)T^{2}
89 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{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

−9.830355363171089486992677814638, −9.488446231581033010177214032296, −8.825289050556513985928721813709, −7.893642125024167756872176641485, −7.45453305709658081889849087604, −5.87660358359649099803992227486, −4.98344910340118423411089077203, −3.79379137017845518994948433391, −2.88382182842019981485235886192, −1.37550894749488483039028189351, 1.25831427561184842027981259474, 2.73920380934427429150708416734, 3.24423136135569436242754514441, 5.25744728013731641641656664934, 6.40000431649447318163284165443, 7.15298692765118034563167672362, 7.45508604780163789903591873857, 8.545308367296151959735496569566, 9.174662474852540780993993240237, 10.09463205852604637222051790477

Graph of the ZZ-function along the critical line