Properties

Label 2-720-80.29-c1-0-25
Degree $2$
Conductor $720$
Sign $0.501 - 0.865i$
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  + (1.39 + 0.258i)2-s + (1.86 + 0.719i)4-s + (−0.404 + 2.19i)5-s + 1.81·7-s + (2.40 + 1.48i)8-s + (−1.13 + 2.95i)10-s + (0.331 − 0.331i)11-s + (−0.0310 + 0.0310i)13-s + (2.52 + 0.469i)14-s + (2.96 + 2.68i)16-s + 1.00i·17-s + (−2.08 − 2.08i)19-s + (−2.33 + 3.81i)20-s + (0.547 − 0.375i)22-s − 6.22·23-s + ⋯
L(s)  = 1  + (0.983 + 0.183i)2-s + (0.933 + 0.359i)4-s + (−0.180 + 0.983i)5-s + 0.686·7-s + (0.851 + 0.524i)8-s + (−0.357 + 0.933i)10-s + (0.100 − 0.100i)11-s + (−0.00860 + 0.00860i)13-s + (0.674 + 0.125i)14-s + (0.740 + 0.671i)16-s + 0.243i·17-s + (−0.477 − 0.477i)19-s + (−0.522 + 0.852i)20-s + (0.116 − 0.0800i)22-s − 1.29·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.501 - 0.865i$
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: \((2,\ 720,\ (\ :1/2),\ 0.501 - 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.52445 + 1.45470i\)
\(L(\frac12)\) \(\approx\) \(2.52445 + 1.45470i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (0.404 - 2.19i)T \)
good7 \( 1 - 1.81T + 7T^{2} \)
11 \( 1 + (-0.331 + 0.331i)T - 11iT^{2} \)
13 \( 1 + (0.0310 - 0.0310i)T - 13iT^{2} \)
17 \( 1 - 1.00iT - 17T^{2} \)
19 \( 1 + (2.08 + 2.08i)T + 19iT^{2} \)
23 \( 1 + 6.22T + 23T^{2} \)
29 \( 1 + (-6.28 - 6.28i)T + 29iT^{2} \)
31 \( 1 - 7.11T + 31T^{2} \)
37 \( 1 + (0.0723 + 0.0723i)T + 37iT^{2} \)
41 \( 1 + 3.06iT - 41T^{2} \)
43 \( 1 + (-3.78 - 3.78i)T + 43iT^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + (7.04 + 7.04i)T + 53iT^{2} \)
59 \( 1 + (-6.68 + 6.68i)T - 59iT^{2} \)
61 \( 1 + (-2.89 - 2.89i)T + 61iT^{2} \)
67 \( 1 + (0.150 - 0.150i)T - 67iT^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 + (5.48 - 5.48i)T - 83iT^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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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

−10.71176672507525405505096656900, −10.08090821523507638337446741386, −8.486290236334446011518987790084, −7.81994726312833449227059025978, −6.80079738594378858029571918575, −6.22051546998804592409020885728, −5.04589391217134455694282453755, −4.12038124804482606717221585255, −3.08211237707809972396794917866, −1.98184105447955546475241000021, 1.27102011555973442294806038860, 2.53791466990947282963088643701, 4.12518949767957865244503506960, 4.57242359879481552979979574517, 5.63619252244577107779459608934, 6.44476776315851838655948500134, 7.79371183697181418798798956449, 8.278912810493984137035371597943, 9.589678380153414920813000828120, 10.37396001162481204234921145397

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