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Properties

Label 2-760-760.189-c0-0-4
Degree 22
Conductor 760760
Sign 0.923−0.382i0.923 - 0.382i
Analytic cond. 0.3792890.379289
Root an. cond. 0.6158640.615864
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 + 0.765i·3-s + (−0.707 − 0.707i)4-s − i·5-s + (−0.707 − 0.292i)6-s + (0.923 − 0.382i)8-s + 0.414·9-s + (0.923 + 0.382i)10-s − 1.41i·11-s + (0.541 − 0.541i)12-s − 1.84i·13-s + 0.765·15-s + i·16-s + (−0.158 + 0.382i)18-s + i·19-s + (−0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + 0.765i·3-s + (−0.707 − 0.707i)4-s − i·5-s + (−0.707 − 0.292i)6-s + (0.923 − 0.382i)8-s + 0.414·9-s + (0.923 + 0.382i)10-s − 1.41i·11-s + (0.541 − 0.541i)12-s − 1.84i·13-s + 0.765·15-s + i·16-s + (−0.158 + 0.382i)18-s + i·19-s + (−0.707 + 0.707i)20-s + ⋯

Functional equation

Λ(s)=(760s/2 ΓC(s) L(s)=((0.923−0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(760s/2 ΓC(s) L(s)=((0.923−0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 760760    =    23⋅5⋅192^{3} \cdot 5 \cdot 19
Sign: 0.923−0.382i0.923 - 0.382i
Analytic conductor: 0.3792890.379289
Root analytic conductor: 0.6158640.615864
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ760(189,⋅)\chi_{760} (189, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 760, ( :0), 0.923−0.382i)(2,\ 760,\ (\ :0),\ 0.923 - 0.382i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.76523159860.7652315986
L(12)L(\frac12) ≈\approx 0.76523159860.7652315986
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
5 1+iT 1 + iT
19 1−iT 1 - iT
good3 1−0.765iT−T2 1 - 0.765iT - T^{2}
7 1−T2 1 - T^{2}
11 1+1.41iT−T2 1 + 1.41iT - T^{2}
13 1+1.84iT−T2 1 + 1.84iT - T^{2}
17 1−T2 1 - T^{2}
23 1−T2 1 - T^{2}
29 1+T2 1 + T^{2}
31 1−T2 1 - T^{2}
37 1−0.765iT−T2 1 - 0.765iT - T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1−T2 1 - T^{2}
53 1+0.765iT−T2 1 + 0.765iT - T^{2}
59 1+T2 1 + T^{2}
61 1−1.41iT−T2 1 - 1.41iT - T^{2}
67 1−1.84iT−T2 1 - 1.84iT - T^{2}
71 1−T2 1 - T^{2}
73 1−T2 1 - T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1−T2 1 - T^{2}
97 1+1.84T+T2 1 + 1.84T + 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

−10.22467616710998489542562032798, −9.774525656887856241351642908736, −8.592753052261509320041223596891, −8.327613216072979561010171038131, −7.32190560437348964866597178948, −5.81730222421585674546816566393, −5.51542614978338252233282019947, −4.43419599092851498938453175301, −3.44280843102116869893743109298, −1.04323228490519563664463447560, 1.78344252486711102387075959893, 2.42085619131412664265747652184, 3.92515368015775189451262716555, 4.71721524418332331860155676176, 6.55009617606950544689008705295, 7.12342606537913083977318859501, 7.73446317341803541229733829064, 9.118886733049526694471597905605, 9.645858487647965762154122222754, 10.55044517793837874624863700973

Graph of the ZZ-function along the critical line