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Properties

Label 2-720-144.11-c1-0-12
Degree $2$
Conductor $720$
Sign $-0.489 - 0.872i$
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  + (−1.12 + 0.858i)2-s + (−1.61 − 0.631i)3-s + (0.525 − 1.92i)4-s + (0.965 + 0.258i)5-s + (2.35 − 0.675i)6-s + (−1.12 + 1.95i)7-s + (1.06 + 2.61i)8-s + (2.20 + 2.03i)9-s + (−1.30 + 0.538i)10-s + (1.28 − 0.344i)11-s + (−2.06 + 2.78i)12-s + (−3.50 − 0.940i)13-s + (−0.409 − 3.15i)14-s + (−1.39 − 1.02i)15-s + (−3.44 − 2.02i)16-s − 6.51i·17-s + ⋯
L(s)  = 1  + (−0.794 + 0.607i)2-s + (−0.931 − 0.364i)3-s + (0.262 − 0.964i)4-s + (0.431 + 0.115i)5-s + (0.961 − 0.275i)6-s + (−0.425 + 0.737i)7-s + (0.377 + 0.926i)8-s + (0.734 + 0.678i)9-s + (−0.413 + 0.170i)10-s + (0.387 − 0.103i)11-s + (−0.596 + 0.802i)12-s + (−0.972 − 0.260i)13-s + (−0.109 − 0.844i)14-s + (−0.360 − 0.265i)15-s + (−0.862 − 0.506i)16-s − 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.489 - 0.872i$
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: \((2,\ 720,\ (\ :1/2),\ -0.489 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.264619 + 0.451876i\)
\(L(\frac12)\) \(\approx\) \(0.264619 + 0.451876i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 - 0.858i)T \)
3 \( 1 + (1.61 + 0.631i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
good7 \( 1 + (1.12 - 1.95i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.28 + 0.344i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.50 + 0.940i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 6.51iT - 17T^{2} \)
19 \( 1 + (-5.35 - 5.35i)T + 19iT^{2} \)
23 \( 1 + (3.23 - 1.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.22 + 0.328i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (7.37 - 4.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.77 - 2.77i)T + 37iT^{2} \)
41 \( 1 + (-0.620 - 1.07i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.74 - 10.2i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.42 - 7.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.816 + 0.816i)T - 53iT^{2} \)
59 \( 1 + (2.90 - 10.8i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.541 - 2.02i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.171 + 0.639i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 13.2iT - 71T^{2} \)
73 \( 1 - 14.2iT - 73T^{2} \)
79 \( 1 + (14.1 + 8.18i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.780 + 2.91i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 0.500T + 89T^{2} \)
97 \( 1 + (5.12 - 8.86i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.46987286797120568851031111363, −9.666978033850601869035716067089, −9.273242337911539123658297202914, −7.79750072742596727124289779644, −7.27999657231104889566436701701, −6.24063101569106070435225488020, −5.63096221617014636262740602268, −4.86207824144330106284096320806, −2.73592817446270620506819062478, −1.32869545660607109195191542562, 0.42632114043922446556289002857, 1.92617166389367940888888415761, 3.57465484558753096111132529962, 4.41869950386312558217375824502, 5.69234908506858570758779097288, 6.83024702888733424590761872630, 7.34934904997526259372229005706, 8.715956049549037120982332002951, 9.662909113278035406768447090903, 10.02538203266465885185480098528

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