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Properties

Label 2-240-240.149-c0-0-0
Degree 22
Conductor 240240
Sign 0.923+0.382i0.923 + 0.382i
Analytic cond. 0.1197750.119775
Root an. cond. 0.3460860.346086
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)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + 1.00·10-s + (0.707 + 0.707i)12-s − 1.00·15-s − 1.00·16-s − 1.41·17-s + (−0.707 − 0.707i)18-s + (−1 + i)19-s + (0.707 − 0.707i)20-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + 1.00·10-s + (0.707 + 0.707i)12-s − 1.00·15-s − 1.00·16-s − 1.41·17-s + (−0.707 − 0.707i)18-s + (−1 + i)19-s + (0.707 − 0.707i)20-s + ⋯

Functional equation

Λ(s)=(240s/2 ΓC(s) L(s)=((0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(240s/2 ΓC(s) L(s)=((0.923+0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 240240    =    24⋅3⋅52^{4} \cdot 3 \cdot 5
Sign: 0.923+0.382i0.923 + 0.382i
Analytic conductor: 0.1197750.119775
Root analytic conductor: 0.3460860.346086
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ240(149,⋅)\chi_{240} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 240, ( :0), 0.923+0.382i)(2,\ 240,\ (\ :0),\ 0.923 + 0.382i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.87666715030.8766671503
L(12)L(\frac12) ≈\approx 0.87666715030.8766671503
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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
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

−12.26721417849261293596645460060, −11.11131220770607798226918165643, −10.64097464556387682232035748743, −9.869253903909641976622981782502, −8.841082183980231248155593852572, −6.64680116357307140519216654282, −6.10824185492806981195453492813, −4.88125479089220107901179134265, −3.82307167587796902320542056164, −2.30617234132287283414117722721, 2.24575443364944395837814610849, 4.41427234721809658050812687514, 5.34247706028749470396480426010, 6.28632739387403651003711591542, 7.10643469094892591735622019280, 8.332017620077728990333548048250, 9.243675452755684177654724180201, 10.84841673103753683404825725208, 11.73527978311156713498965175251, 12.73502049729843391068627570637

Graph of the ZZ-function along the critical line