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Properties

Label 2-720-16.5-c1-0-21
Degree $2$
Conductor $720$
Sign $0.709 + 0.704i$
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 + 0.139i)2-s + (1.96 − 0.393i)4-s + (−0.707 − 0.707i)5-s − 0.982i·7-s + (−2.70 + 0.828i)8-s + (1.09 + 0.896i)10-s + (1.62 + 1.62i)11-s + (−0.690 + 0.690i)13-s + (0.137 + 1.38i)14-s + (3.68 − 1.54i)16-s + 2.19·17-s + (1.92 − 1.92i)19-s + (−1.66 − 1.10i)20-s + (−2.51 − 2.06i)22-s − 2.01i·23-s + ⋯
L(s)  = 1  + (−0.995 + 0.0989i)2-s + (0.980 − 0.196i)4-s + (−0.316 − 0.316i)5-s − 0.371i·7-s + (−0.956 + 0.292i)8-s + (0.345 + 0.283i)10-s + (0.490 + 0.490i)11-s + (−0.191 + 0.191i)13-s + (0.0367 + 0.369i)14-s + (0.922 − 0.386i)16-s + 0.532·17-s + (0.441 − 0.441i)19-s + (−0.372 − 0.247i)20-s + (−0.536 − 0.439i)22-s − 0.420i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.709 + 0.704i$
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: \((2,\ 720,\ (\ :1/2),\ 0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.831570 - 0.342610i\)
\(L(\frac12)\) \(\approx\) \(0.831570 - 0.342610i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.139i)T \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + 0.982iT - 7T^{2} \)
11 \( 1 + (-1.62 - 1.62i)T + 11iT^{2} \)
13 \( 1 + (0.690 - 0.690i)T - 13iT^{2} \)
17 \( 1 - 2.19T + 17T^{2} \)
19 \( 1 + (-1.92 + 1.92i)T - 19iT^{2} \)
23 \( 1 + 2.01iT - 23T^{2} \)
29 \( 1 + (-5.27 + 5.27i)T - 29iT^{2} \)
31 \( 1 - 0.435T + 31T^{2} \)
37 \( 1 + (5.79 + 5.79i)T + 37iT^{2} \)
41 \( 1 + 3.93iT - 41T^{2} \)
43 \( 1 + (0.507 + 0.507i)T + 43iT^{2} \)
47 \( 1 - 9.21T + 47T^{2} \)
53 \( 1 + (6.29 + 6.29i)T + 53iT^{2} \)
59 \( 1 + (-5.67 - 5.67i)T + 59iT^{2} \)
61 \( 1 + (3.60 - 3.60i)T - 61iT^{2} \)
67 \( 1 + (-4.53 + 4.53i)T - 67iT^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + 9.24iT - 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 + (-0.683 + 0.683i)T - 83iT^{2} \)
89 \( 1 + 5.44iT - 89T^{2} \)
97 \( 1 - 5.54T + 97T^{2} \)
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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

−10.21497354206758767743807831427, −9.374210717416287518030522796859, −8.684015926839500939412098692330, −7.69094731635650886390124704539, −7.08508034080111052302353701280, −6.10660637401405119525201061074, −4.89520897181029779280959735370, −3.65243284044416330324308818817, −2.22253719855650606616391252271, −0.76306117217999120351045422047, 1.21307765760534094547451049113, 2.75471591978837740127641500618, 3.65857203974905980183136588273, 5.32239346638386385458175339742, 6.33530834069295499570501021232, 7.17823358592651477014082833394, 8.073881284638365696282677934177, 8.764976721752270999197193603865, 9.673539466815841268328058630591, 10.41640175668965220942938545268

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