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Properties

Label 2-960-240.149-c0-0-0
Degree 22
Conductor 960960
Sign 0.923+0.382i0.923 + 0.382i
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  + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)5-s − 1.00i·9-s + 1.00·15-s + 1.41·17-s + (1 − i)19-s − 1.41i·23-s + 1.00i·25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)45-s + 1.41·47-s − 49-s + (−1.00 + 1.00i)51-s + 1.41i·57-s + (1 − i)61-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (−0.707 − 0.707i)5-s − 1.00i·9-s + 1.00·15-s + 1.41·17-s + (1 − i)19-s − 1.41i·23-s + 1.00i·25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)45-s + 1.41·47-s − 49-s + (−1.00 + 1.00i)51-s + 1.41i·57-s + (1 − i)61-s + ⋯

Functional equation

Λ(s)=(960s/2 ΓC(s) L(s)=((0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(960s/2 ΓC(s) L(s)=((0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 960960    =    26⋅3⋅52^{6} \cdot 3 \cdot 5
Sign: 0.923+0.382i0.923 + 0.382i
Analytic conductor: 0.4791020.479102
Root analytic conductor: 0.6921720.692172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ960(689,⋅)\chi_{960} (689, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 960, ( :0), 0.923+0.382i)(2,\ 960,\ (\ :0),\ 0.923 + 0.382i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.71565923480.7156592348
L(12)L(\frac12) ≈\approx 0.71565923480.7156592348
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
3 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
5 1+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
good7 1+T2 1 + T^{2}
11 1−iT2 1 - iT^{2}
13 1+iT2 1 + iT^{2}
17 1−1.41T+T2 1 - 1.41T + T^{2}
19 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
23 1+1.41iT−T2 1 + 1.41iT - T^{2}
29 1+iT2 1 + iT^{2}
31 1+T2 1 + T^{2}
37 1−iT2 1 - iT^{2}
41 1+T2 1 + T^{2}
43 1−iT2 1 - iT^{2}
47 1−1.41T+T2 1 - 1.41T + T^{2}
53 1+iT2 1 + iT^{2}
59 1−iT2 1 - iT^{2}
61 1+(−1+i)T−iT2 1 + (-1 + i)T - iT^{2}
67 1+iT2 1 + iT^{2}
71 1+T2 1 + T^{2}
73 1+T2 1 + T^{2}
79 1+2T+T2 1 + 2T + T^{2}
83 1−iT2 1 - iT^{2}
89 1+T2 1 + 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

−10.17185099405675579405742713151, −9.412012125410053695396653930178, −8.635805100144743904129373147934, −7.69185524582873840404655483806, −6.75078818536465880981433340165, −5.59723686904818904174306294797, −4.94039792207487197577294687028, −4.07843222763765551901920781336, −3.07027421732028272537879621192, −0.894775071984483033248944915766, 1.37174117917484634184311467975, 2.95074215193585422544786187958, 3.93439942715813729771767355662, 5.34513621981013508106646170895, 5.93501165329974293263810987650, 7.09789416649322380977365357529, 7.58000752147327875751370612629, 8.255737972740826183164743737639, 9.698155205188167412355580046299, 10.36558232986173764903152328891

Graph of the ZZ-function along the critical line