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Properties

Label 2-720-60.23-c0-0-0
Degree 22
Conductor 720720
Sign 0.662−0.749i0.662 - 0.749i
Analytic cond. 0.3593260.359326
Root an. cond. 0.5994380.599438
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)5-s + (1 + i)13-s + (1.41 + 1.41i)17-s − 1.00i·25-s − 1.41·29-s + (−1 + i)37-s − 1.41i·41-s − i·49-s + (1.41 − 1.41i)53-s − 1.41·65-s + (−1 − i)73-s − 2.00·85-s + 1.41·89-s + (−1 + i)97-s − 1.41i·101-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)5-s + (1 + i)13-s + (1.41 + 1.41i)17-s − 1.00i·25-s − 1.41·29-s + (−1 + i)37-s − 1.41i·41-s − i·49-s + (1.41 − 1.41i)53-s − 1.41·65-s + (−1 − i)73-s − 2.00·85-s + 1.41·89-s + (−1 + i)97-s − 1.41i·101-s + ⋯

Functional equation

Λ(s)=(720s/2 ΓC(s) L(s)=((0.662−0.749i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(720s/2 ΓC(s) L(s)=((0.662−0.749i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 720720    =    24⋅32⋅52^{4} \cdot 3^{2} \cdot 5
Sign: 0.662−0.749i0.662 - 0.749i
Analytic conductor: 0.3593260.359326
Root analytic conductor: 0.5994380.599438
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ720(143,⋅)\chi_{720} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 720, ( :0), 0.662−0.749i)(2,\ 720,\ (\ :0),\ 0.662 - 0.749i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.87444772390.8744477239
L(12)L(\frac12) ≈\approx 0.87444772390.8744477239
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 1
5 1+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good7 1+iT2 1 + iT^{2}
11 1+T2 1 + T^{2}
13 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
17 1+(−1.41−1.41i)T+iT2 1 + (-1.41 - 1.41i)T + iT^{2}
19 1+T2 1 + T^{2}
23 1+iT2 1 + iT^{2}
29 1+1.41T+T2 1 + 1.41T + T^{2}
31 1−T2 1 - T^{2}
37 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
41 1+1.41iT−T2 1 + 1.41iT - T^{2}
43 1−iT2 1 - iT^{2}
47 1−iT2 1 - iT^{2}
53 1+(−1.41+1.41i)T−iT2 1 + (-1.41 + 1.41i)T - iT^{2}
59 1−T2 1 - T^{2}
61 1+T2 1 + T^{2}
67 1+iT2 1 + iT^{2}
71 1+T2 1 + T^{2}
73 1+(1+i)T+iT2 1 + (1 + i)T + iT^{2}
79 1+T2 1 + T^{2}
83 1+iT2 1 + iT^{2}
89 1−1.41T+T2 1 - 1.41T + T^{2}
97 1+(1−i)T−iT2 1 + (1 - i)T - iT^{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

−10.69586414033327449488954340451, −10.09526613516142092490867168529, −8.875895662919085980133728579828, −8.158999394857400877990101158500, −7.24121539184512390265557339016, −6.41368655815196284656496659776, −5.46105895518190051972033628609, −3.94681818133471158596379612285, −3.48464899534862767580549396055, −1.79867177698340179480444143221, 1.09226915015899176819216067128, 3.04288437003052259917622483598, 3.95107740013189464419388771609, 5.17495173136970089075037723889, 5.82121470846177135671677441346, 7.31779022604041152949830528562, 7.83549547061645042121765394623, 8.785891869991726383636566755147, 9.534181708363558749196133230669, 10.56789003391955680103010733342

Graph of the ZZ-function along the critical line