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Properties

Label 2-720-80.69-c1-0-22
Degree $2$
Conductor $720$
Sign $0.916 + 0.399i$
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  + (−0.903 − 1.08i)2-s + (−0.368 + 1.96i)4-s + (2.09 − 0.770i)5-s + 3.05·7-s + (2.47 − 1.37i)8-s + (−2.73 − 1.58i)10-s + (1.80 + 1.80i)11-s + (2.47 + 2.47i)13-s + (−2.75 − 3.31i)14-s + (−3.72 − 1.44i)16-s + 3.66i·17-s + (−2.31 + 2.31i)19-s + (0.741 + 4.41i)20-s + (0.334 − 3.60i)22-s − 4.86·23-s + ⋯
L(s)  = 1  + (−0.638 − 0.769i)2-s + (−0.184 + 0.982i)4-s + (0.938 − 0.344i)5-s + 1.15·7-s + (0.873 − 0.486i)8-s + (−0.864 − 0.502i)10-s + (0.545 + 0.545i)11-s + (0.685 + 0.685i)13-s + (−0.736 − 0.887i)14-s + (−0.932 − 0.361i)16-s + 0.889i·17-s + (−0.531 + 0.531i)19-s + (0.165 + 0.986i)20-s + (0.0713 − 0.768i)22-s − 1.01·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.916 + 0.399i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.916 + 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45958 - 0.304539i\)
\(L(\frac12)\) \(\approx\) \(1.45958 - 0.304539i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.903 + 1.08i)T \)
3 \( 1 \)
5 \( 1 + (-2.09 + 0.770i)T \)
good7 \( 1 - 3.05T + 7T^{2} \)
11 \( 1 + (-1.80 - 1.80i)T + 11iT^{2} \)
13 \( 1 + (-2.47 - 2.47i)T + 13iT^{2} \)
17 \( 1 - 3.66iT - 17T^{2} \)
19 \( 1 + (2.31 - 2.31i)T - 19iT^{2} \)
23 \( 1 + 4.86T + 23T^{2} \)
29 \( 1 + (4.74 - 4.74i)T - 29iT^{2} \)
31 \( 1 - 1.86T + 31T^{2} \)
37 \( 1 + (-5.40 + 5.40i)T - 37iT^{2} \)
41 \( 1 - 6.47iT - 41T^{2} \)
43 \( 1 + (4.19 - 4.19i)T - 43iT^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 + (-9.99 + 9.99i)T - 53iT^{2} \)
59 \( 1 + (-2.47 - 2.47i)T + 59iT^{2} \)
61 \( 1 + (-8.01 + 8.01i)T - 61iT^{2} \)
67 \( 1 + (8.60 + 8.60i)T + 67iT^{2} \)
71 \( 1 - 6.63iT - 71T^{2} \)
73 \( 1 - 2.70T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + (3.65 + 3.65i)T + 83iT^{2} \)
89 \( 1 + 13.1iT - 89T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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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

−10.26504243027507787688445022210, −9.614368074457860877960940560566, −8.634598457116550747515874264404, −8.213486850902601953472167059901, −6.97248281813378616236060807377, −5.90771988245784097554526602024, −4.64031164670971847789371939416, −3.82127732880201331274900212861, −2.00644438240468454518997545125, −1.54121270028190164350798899055, 1.14632812441912611218284827156, 2.41961891670945598454894169780, 4.26617788460014428216427785946, 5.44088018743843400904590619391, 6.01280864025049796019700909247, 6.99095750316759589075973381004, 7.960148527318181477934379976164, 8.680989745979063760130498973540, 9.479455385579266166456527089726, 10.36746598865618831708460730512

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