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Properties

Label 2-720-80.29-c1-0-9
Degree $2$
Conductor $720$
Sign $-0.281 - 0.959i$
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  + (−1.40 − 0.112i)2-s + (1.97 + 0.316i)4-s + (0.466 + 2.18i)5-s − 1.00·7-s + (−2.74 − 0.667i)8-s + (−0.413 − 3.13i)10-s + (1.89 − 1.89i)11-s + (−2.65 + 2.65i)13-s + (1.40 + 0.112i)14-s + (3.80 + 1.24i)16-s + 1.73i·17-s + (5.33 + 5.33i)19-s + (0.230 + 4.46i)20-s + (−2.88 + 2.45i)22-s + 0.160·23-s + ⋯
L(s)  = 1  + (−0.996 − 0.0792i)2-s + (0.987 + 0.158i)4-s + (0.208 + 0.977i)5-s − 0.378·7-s + (−0.971 − 0.235i)8-s + (−0.130 − 0.991i)10-s + (0.571 − 0.571i)11-s + (−0.737 + 0.737i)13-s + (0.376 + 0.0299i)14-s + (0.950 + 0.312i)16-s + 0.421i·17-s + (1.22 + 1.22i)19-s + (0.0516 + 0.998i)20-s + (−0.614 + 0.524i)22-s + 0.0334·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.281 - 0.959i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.281 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.452161 + 0.604036i\)
\(L(\frac12)\) \(\approx\) \(0.452161 + 0.604036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.112i)T \)
3 \( 1 \)
5 \( 1 + (-0.466 - 2.18i)T \)
good7 \( 1 + 1.00T + 7T^{2} \)
11 \( 1 + (-1.89 + 1.89i)T - 11iT^{2} \)
13 \( 1 + (2.65 - 2.65i)T - 13iT^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
19 \( 1 + (-5.33 - 5.33i)T + 19iT^{2} \)
23 \( 1 - 0.160T + 23T^{2} \)
29 \( 1 + (2.70 + 2.70i)T + 29iT^{2} \)
31 \( 1 + 4.64T + 31T^{2} \)
37 \( 1 + (5.35 + 5.35i)T + 37iT^{2} \)
41 \( 1 - 9.89iT - 41T^{2} \)
43 \( 1 + (-7.23 - 7.23i)T + 43iT^{2} \)
47 \( 1 - 4.79iT - 47T^{2} \)
53 \( 1 + (-3.44 - 3.44i)T + 53iT^{2} \)
59 \( 1 + (0.101 - 0.101i)T - 59iT^{2} \)
61 \( 1 + (6.01 + 6.01i)T + 61iT^{2} \)
67 \( 1 + (9.04 - 9.04i)T - 67iT^{2} \)
71 \( 1 - 4.60iT - 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + 5.73T + 79T^{2} \)
83 \( 1 + (-2.04 + 2.04i)T - 83iT^{2} \)
89 \( 1 - 15.0iT - 89T^{2} \)
97 \( 1 + 3.84iT - 97T^{2} \)
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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

−10.55788582475953981712460160872, −9.634796736740811252087316380288, −9.300787461956685348938002744643, −7.971506828217266199973462850970, −7.31627356156607784424803363334, −6.41352647217032213818662083328, −5.73894017712883584460262046214, −3.81615231821328760643256736782, −2.89357016191008662756392012539, −1.61869310657269855752029320069, 0.53638222185186715523460097441, 1.96477716002239382729258286283, 3.31266075502364200633100499453, 4.93403880512060740369768555230, 5.67926119510648034741438151240, 7.02181427160978296707172477869, 7.48096370220204096984903061369, 8.720242293430943010624525412415, 9.271736004049352734943921846748, 9.871656218405515385058006844696

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