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Properties

Label 2-720-16.5-c1-0-13
Degree $2$
Conductor $720$
Sign $0.874 - 0.485i$
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  + (−1.32 − 0.503i)2-s + (1.49 + 1.33i)4-s + (0.707 + 0.707i)5-s + 2.69i·7-s + (−1.30 − 2.51i)8-s + (−0.578 − 1.29i)10-s + (−2.72 − 2.72i)11-s + (1.82 − 1.82i)13-s + (1.35 − 3.56i)14-s + (0.455 + 3.97i)16-s + 7.33·17-s + (3.62 − 3.62i)19-s + (0.114 + 1.99i)20-s + (2.23 + 4.97i)22-s + 8.95i·23-s + ⋯
L(s)  = 1  + (−0.934 − 0.356i)2-s + (0.746 + 0.665i)4-s + (0.316 + 0.316i)5-s + 1.01i·7-s + (−0.460 − 0.887i)8-s + (−0.182 − 0.408i)10-s + (−0.822 − 0.822i)11-s + (0.506 − 0.506i)13-s + (0.362 − 0.951i)14-s + (0.113 + 0.993i)16-s + 1.77·17-s + (0.831 − 0.831i)19-s + (0.0254 + 0.446i)20-s + (0.475 + 1.06i)22-s + 1.86i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.874 - 0.485i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.874 - 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01287 + 0.262297i\)
\(L(\frac12)\) \(\approx\) \(1.01287 + 0.262297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.503i)T \)
3 \( 1 \)
5 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 - 2.69iT - 7T^{2} \)
11 \( 1 + (2.72 + 2.72i)T + 11iT^{2} \)
13 \( 1 + (-1.82 + 1.82i)T - 13iT^{2} \)
17 \( 1 - 7.33T + 17T^{2} \)
19 \( 1 + (-3.62 + 3.62i)T - 19iT^{2} \)
23 \( 1 - 8.95iT - 23T^{2} \)
29 \( 1 + (2.84 - 2.84i)T - 29iT^{2} \)
31 \( 1 + 3.37T + 31T^{2} \)
37 \( 1 + (0.190 + 0.190i)T + 37iT^{2} \)
41 \( 1 - 7.67iT - 41T^{2} \)
43 \( 1 + (-7.98 - 7.98i)T + 43iT^{2} \)
47 \( 1 - 1.31T + 47T^{2} \)
53 \( 1 + (-6.71 - 6.71i)T + 53iT^{2} \)
59 \( 1 + (1.01 + 1.01i)T + 59iT^{2} \)
61 \( 1 + (-2.38 + 2.38i)T - 61iT^{2} \)
67 \( 1 + (7.22 - 7.22i)T - 67iT^{2} \)
71 \( 1 - 2.28iT - 71T^{2} \)
73 \( 1 + 1.31iT - 73T^{2} \)
79 \( 1 + 2.59T + 79T^{2} \)
83 \( 1 + (-5.36 + 5.36i)T - 83iT^{2} \)
89 \( 1 + 14.8iT - 89T^{2} \)
97 \( 1 - 0.694T + 97T^{2} \)
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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

−10.40797746011883543809321132333, −9.554019637756116010251544417300, −8.949301304465743726472474032550, −7.894936455434044498800250220322, −7.39177796969552291274347739021, −5.85992594843337069744603958442, −5.51998372743422647461357242249, −3.34723960805241742896811026456, −2.82038482668236592155461112416, −1.26652640414397549154034285868, 0.867138565895150028694800821094, 2.18889045957673829820896266224, 3.80692172752873034875121118928, 5.15827294041491528882601523168, 5.98363216764708722810464515059, 7.19228862393691854836390896834, 7.64643517094541528523319380774, 8.587943859641747538615800040381, 9.585051746173051698529382933400, 10.29691289295511601908130860497

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