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Properties

Label 2-720-144.11-c1-0-22
Degree $2$
Conductor $720$
Sign $-0.842 - 0.537i$
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  + (−1.26 + 0.636i)2-s + (0.418 + 1.68i)3-s + (1.19 − 1.60i)4-s + (−0.965 − 0.258i)5-s + (−1.59 − 1.85i)6-s + (−0.905 + 1.56i)7-s + (−0.480 + 2.78i)8-s + (−2.64 + 1.40i)9-s + (1.38 − 0.287i)10-s + (4.47 − 1.19i)11-s + (3.19 + 1.32i)12-s + (5.71 + 1.53i)13-s + (0.145 − 2.55i)14-s + (0.0302 − 1.73i)15-s + (−1.16 − 3.82i)16-s + 3.09i·17-s + ⋯
L(s)  = 1  + (−0.893 + 0.449i)2-s + (0.241 + 0.970i)3-s + (0.595 − 0.803i)4-s + (−0.431 − 0.115i)5-s + (−0.652 − 0.757i)6-s + (−0.342 + 0.592i)7-s + (−0.169 + 0.985i)8-s + (−0.882 + 0.469i)9-s + (0.437 − 0.0909i)10-s + (1.34 − 0.361i)11-s + (0.923 + 0.383i)12-s + (1.58 + 0.424i)13-s + (0.0389 − 0.683i)14-s + (0.00781 − 0.447i)15-s + (−0.291 − 0.956i)16-s + 0.750i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.842 - 0.537i$
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: \((2,\ 720,\ (\ :1/2),\ -0.842 - 0.537i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.251362 + 0.861168i\)
\(L(\frac12)\) \(\approx\) \(0.251362 + 0.861168i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 - 0.636i)T \)
3 \( 1 + (-0.418 - 1.68i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
good7 \( 1 + (0.905 - 1.56i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.47 + 1.19i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-5.71 - 1.53i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 3.09iT - 17T^{2} \)
19 \( 1 + (-0.900 - 0.900i)T + 19iT^{2} \)
23 \( 1 + (4.95 - 2.85i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (10.0 - 2.69i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-2.24 + 1.29i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.80 - 3.80i)T + 37iT^{2} \)
41 \( 1 + (-0.0338 - 0.0586i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.38 + 8.91i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.84 - 3.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.66 - 4.66i)T - 53iT^{2} \)
59 \( 1 + (1.50 - 5.62i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.72 - 6.42i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.93 - 10.9i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.16iT - 71T^{2} \)
73 \( 1 - 4.15iT - 73T^{2} \)
79 \( 1 + (14.7 + 8.49i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.11 + 7.89i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 9.02T + 89T^{2} \)
97 \( 1 + (-5.98 + 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.60547122382352762425726635399, −9.663280995855011623943048722466, −8.883764547363645043598570476992, −8.614487575354990781189944658654, −7.51238881396954343172976335391, −6.10773799485486934435037740803, −5.83681480293913384956132558057, −4.19831585638223900204546663892, −3.38091466344548783216888784937, −1.62061698298665133229728507137, 0.64047718063513402007456743197, 1.81198188385805044277531147677, 3.28945849869317996762230461849, 3.98541136546978722120528356471, 6.13521960866306064315032829362, 6.74652534159196854197546189598, 7.59528831512608333621162595983, 8.303862184779596401162412349387, 9.159384912553531230575776179161, 9.901156280232017020204966990783

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