↑

Properties

Label 2-720-144.11-c1-0-37
Degree $2$
Conductor $720$
Sign $-0.673 - 0.739i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.801 + 1.16i)2-s + (1.70 + 0.275i)3-s + (−0.714 + 1.86i)4-s + (0.965 + 0.258i)5-s + (1.04 + 2.21i)6-s + (−1.19 + 2.07i)7-s + (−2.74 + 0.666i)8-s + (2.84 + 0.943i)9-s + (0.473 + 1.33i)10-s + (−2.83 + 0.760i)11-s + (−1.73 + 2.99i)12-s + (−5.79 − 1.55i)13-s + (−3.37 + 0.268i)14-s + (1.58 + 0.709i)15-s + (−2.98 − 2.66i)16-s + 3.09i·17-s + ⋯
L(s)  = 1  + (0.566 + 0.823i)2-s + (0.987 + 0.159i)3-s + (−0.357 + 0.934i)4-s + (0.431 + 0.115i)5-s + (0.428 + 0.903i)6-s + (−0.452 + 0.784i)7-s + (−0.971 + 0.235i)8-s + (0.949 + 0.314i)9-s + (0.149 + 0.421i)10-s + (−0.855 + 0.229i)11-s + (−0.501 + 0.865i)12-s + (−1.60 − 0.430i)13-s + (−0.903 + 0.0717i)14-s + (0.408 + 0.183i)15-s + (−0.745 − 0.667i)16-s + 0.751i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.673 - 0.739i$
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: \((2,\ 720,\ (\ :1/2),\ -0.673 - 0.739i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01516 + 2.29785i\)
\(L(\frac12)\) \(\approx\) \(1.01516 + 2.29785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.801 - 1.16i)T \)
3 \( 1 + (-1.70 - 0.275i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
good7 \( 1 + (1.19 - 2.07i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.83 - 0.760i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (5.79 + 1.55i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 3.09iT - 17T^{2} \)
19 \( 1 + (-4.64 - 4.64i)T + 19iT^{2} \)
23 \( 1 + (-4.71 + 2.72i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.53 + 1.75i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-9.21 + 5.32i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.146 + 0.146i)T + 37iT^{2} \)
41 \( 1 + (0.643 + 1.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.172 - 0.645i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.18 + 7.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.07 - 4.07i)T - 53iT^{2} \)
59 \( 1 + (2.86 - 10.6i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.48 + 9.27i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.86 + 10.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 + 1.68iT - 73T^{2} \)
79 \( 1 + (10.7 + 6.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.98 + 11.1i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 9.88T + 89T^{2} \)
97 \( 1 + (-3.42 + 5.93i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.26261842310393588897961763904, −9.795549574403709032049120392119, −8.845485037825375497137049582335, −7.993044648410693267826002313164, −7.38177475622816651997606260031, −6.28017617239659196329668566641, −5.30695988960943627792000741090, −4.47574991542458954933051820226, −3.01289776801120855977554306994, −2.51667541710380492579491769586, 1.00588634608213800252283311729, 2.68049137405635417912228735337, 3.01935422513816493153998316806, 4.57555948451205877737357479524, 5.11384886025319893282077012546, 6.75754228960212962743864615984, 7.31280750083478057129416151582, 8.622037704545848013980289457838, 9.701413654101402401878666472303, 9.810672895931758648092879088020

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