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Properties

Label 2-720-144.11-c1-0-16
Degree $2$
Conductor $720$
Sign $-0.563 + 0.826i$
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  + (0.734 + 1.20i)2-s + (−1.13 + 1.30i)3-s + (−0.921 + 1.77i)4-s + (0.965 + 0.258i)5-s + (−2.41 − 0.411i)6-s + (−2.44 + 4.23i)7-s + (−2.82 + 0.190i)8-s + (−0.422 − 2.97i)9-s + (0.396 + 1.35i)10-s + (−2.73 + 0.733i)11-s + (−1.27 − 3.22i)12-s + (4.66 + 1.24i)13-s + (−6.92 + 0.155i)14-s + (−1.43 + 0.969i)15-s + (−2.30 − 3.27i)16-s − 3.56i·17-s + ⋯
L(s)  = 1  + (0.519 + 0.854i)2-s + (−0.655 + 0.755i)3-s + (−0.460 + 0.887i)4-s + (0.431 + 0.115i)5-s + (−0.985 − 0.167i)6-s + (−0.925 + 1.60i)7-s + (−0.997 + 0.0672i)8-s + (−0.140 − 0.990i)9-s + (0.125 + 0.429i)10-s + (−0.825 + 0.221i)11-s + (−0.368 − 0.929i)12-s + (1.29 + 0.346i)13-s + (−1.84 + 0.0414i)14-s + (−0.370 + 0.250i)15-s + (−0.575 − 0.817i)16-s − 0.865i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.563 + 0.826i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.563 + 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.457076 - 0.864431i\)
\(L(\frac12)\) \(\approx\) \(0.457076 - 0.864431i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.734 - 1.20i)T \)
3 \( 1 + (1.13 - 1.30i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
good7 \( 1 + (2.44 - 4.23i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.73 - 0.733i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-4.66 - 1.24i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 3.56iT - 17T^{2} \)
19 \( 1 + (-0.879 - 0.879i)T + 19iT^{2} \)
23 \( 1 + (-3.76 + 2.17i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.87 - 1.57i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (6.43 - 3.71i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0686 + 0.0686i)T + 37iT^{2} \)
41 \( 1 + (-4.11 - 7.12i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.78 + 6.65i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.0739 + 0.128i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.14 - 2.14i)T - 53iT^{2} \)
59 \( 1 + (-1.59 + 5.94i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.69 - 13.7i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.871 + 3.25i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 5.68iT - 71T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 + (6.04 + 3.48i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.80 + 10.4i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.82T + 89T^{2} \)
97 \( 1 + (8.20 - 14.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.09120808705156037679539139716, −9.899884970088186156591951378825, −9.082185814866452169316929598325, −8.698921539013634405619817293787, −7.14325876457461720993581398728, −6.26301139689443216885835260650, −5.60593951953924017482956269116, −5.05953692144054616574164011007, −3.64221011916608769335619631849, −2.73516638220955433136171751429, 0.46558388853092423212419563204, 1.62317078764131146237625453028, 3.18980778066411726863268106165, 4.11098028884834812622738459440, 5.44803996030939764718628988892, 6.07929622555591107904244700062, 7.00268020557214487388930015302, 8.008895092714325535441003578080, 9.304947636645251339202438783197, 10.23237934158598651367603935187

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