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Properties

Label 2-720-144.11-c1-0-11
Degree $2$
Conductor $720$
Sign $-0.746 - 0.665i$
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 − 0.216i)2-s + (−0.285 + 1.70i)3-s + (1.90 + 0.606i)4-s + (0.965 + 0.258i)5-s + (0.768 − 2.32i)6-s + (0.981 − 1.70i)7-s + (−2.53 − 1.26i)8-s + (−2.83 − 0.974i)9-s + (−1.29 − 0.571i)10-s + (−0.698 + 0.187i)11-s + (−1.57 + 3.08i)12-s + (−4.34 − 1.16i)13-s + (−1.74 + 2.16i)14-s + (−0.717 + 1.57i)15-s + (3.26 + 2.31i)16-s + 7.45i·17-s + ⋯
L(s)  = 1  + (−0.988 − 0.153i)2-s + (−0.164 + 0.986i)3-s + (0.952 + 0.303i)4-s + (0.431 + 0.115i)5-s + (0.313 − 0.949i)6-s + (0.371 − 0.642i)7-s + (−0.895 − 0.445i)8-s + (−0.945 − 0.324i)9-s + (−0.409 − 0.180i)10-s + (−0.210 + 0.0564i)11-s + (−0.455 + 0.890i)12-s + (−1.20 − 0.322i)13-s + (−0.465 + 0.578i)14-s + (−0.185 + 0.407i)15-s + (0.816 + 0.577i)16-s + 1.80i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 - 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.746 - 0.665i$
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: \((2,\ 720,\ (\ :1/2),\ -0.746 - 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.219056 + 0.574659i\)
\(L(\frac12)\) \(\approx\) \(0.219056 + 0.574659i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.216i)T \)
3 \( 1 + (0.285 - 1.70i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
good7 \( 1 + (-0.981 + 1.70i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.698 - 0.187i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (4.34 + 1.16i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 7.45iT - 17T^{2} \)
19 \( 1 + (-3.65 - 3.65i)T + 19iT^{2} \)
23 \( 1 + (5.45 - 3.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.99 - 1.07i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (2.00 - 1.15i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.59 - 1.59i)T + 37iT^{2} \)
41 \( 1 + (-4.94 - 8.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.397 + 1.48i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.89 - 6.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.97 - 6.97i)T - 53iT^{2} \)
59 \( 1 + (-3.13 + 11.6i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.887 - 3.31i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.88 + 7.02i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.140iT - 71T^{2} \)
73 \( 1 - 3.73iT - 73T^{2} \)
79 \( 1 + (4.99 + 2.88i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.83 + 6.85i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 4.97T + 89T^{2} \)
97 \( 1 + (7.01 - 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.53083689606653466300573901908, −9.869593709147028666936716263997, −9.405781773764987629225467421086, −8.095169138432618811712243831674, −7.67526577526811733637362476595, −6.28405836256850876614127283160, −5.50282132597603574662096307896, −4.16894632398045950846564874752, −3.12161069439286800684140459427, −1.67502952234995415535528439375, 0.42364236399558261693873403326, 2.08958682323919874208216007610, 2.66543146611259058260356640902, 5.10505521920602592774789337565, 5.70934409001803868034854489091, 6.94139070280960455344383983780, 7.38074927484897683157650992277, 8.327895526044401286714059429003, 9.232759027433500637349895400323, 9.784222024455377596498494917777

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