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Properties

Label 2-720-240.179-c1-0-6
Degree $2$
Conductor $720$
Sign $-0.999 + 0.0423i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.23 + 0.685i)2-s + (1.05 − 1.69i)4-s + (−0.419 + 2.19i)5-s − 0.263i·7-s + (−0.147 + 2.82i)8-s + (−0.986 − 3.00i)10-s + (−0.913 + 0.913i)11-s + (−1.49 + 1.49i)13-s + (0.180 + 0.326i)14-s + (−1.75 − 3.59i)16-s − 2.90·17-s + (−0.296 + 0.296i)19-s + (3.28 + 3.03i)20-s + (0.503 − 1.75i)22-s + 3.14·23-s + ⋯
L(s)  = 1  + (−0.874 + 0.484i)2-s + (0.529 − 0.848i)4-s + (−0.187 + 0.982i)5-s − 0.0997i·7-s + (−0.0522 + 0.998i)8-s + (−0.312 − 0.950i)10-s + (−0.275 + 0.275i)11-s + (−0.413 + 0.413i)13-s + (0.0483 + 0.0871i)14-s + (−0.438 − 0.898i)16-s − 0.704·17-s + (−0.0679 + 0.0679i)19-s + (0.733 + 0.679i)20-s + (0.107 − 0.374i)22-s + 0.656·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0423i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0423i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.999 + 0.0423i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.999 + 0.0423i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00851469 - 0.402348i\)
\(L(\frac12)\) \(\approx\) \(0.00851469 - 0.402348i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.23 - 0.685i)T \)
3 \( 1 \)
5 \( 1 + (0.419 - 2.19i)T \)
good7 \( 1 + 0.263iT - 7T^{2} \)
11 \( 1 + (0.913 - 0.913i)T - 11iT^{2} \)
13 \( 1 + (1.49 - 1.49i)T - 13iT^{2} \)
17 \( 1 + 2.90T + 17T^{2} \)
19 \( 1 + (0.296 - 0.296i)T - 19iT^{2} \)
23 \( 1 - 3.14T + 23T^{2} \)
29 \( 1 + (2.43 - 2.43i)T - 29iT^{2} \)
31 \( 1 - 2.53iT - 31T^{2} \)
37 \( 1 + (1.53 + 1.53i)T + 37iT^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 + (4.51 - 4.51i)T - 43iT^{2} \)
47 \( 1 + 6.67iT - 47T^{2} \)
53 \( 1 + (2.95 - 2.95i)T - 53iT^{2} \)
59 \( 1 + (9.76 - 9.76i)T - 59iT^{2} \)
61 \( 1 + (1.02 + 1.02i)T + 61iT^{2} \)
67 \( 1 + (-3.67 - 3.67i)T + 67iT^{2} \)
71 \( 1 + 7.85iT - 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 10.7iT - 79T^{2} \)
83 \( 1 + (-3.50 + 3.50i)T - 83iT^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 11.1iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.65216255463385857119450780698, −10.02194551637733504968376032912, −9.100278685500802947634791413181, −8.247275293252594881752281251657, −7.14859686232675493390335325487, −6.89612756698045989796272675114, −5.75925880028867184097770539017, −4.61382802048829589722404796152, −3.07268640999268075733021390202, −1.88269044481232811563533028078, 0.26443989577778019553608888252, 1.78026147115086257737819037082, 3.09830415235240055229941572526, 4.32116880968188456540762115430, 5.39518393519817690147071403168, 6.66546345595331173889781647374, 7.69375879423678138484359129199, 8.404894483999807062713124539077, 9.104203760499289241443196672150, 9.866260145528957810838278071664

Graph of the $Z$-function along the critical line