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Properties

Label 2-720-9.4-c1-0-18
Degree $2$
Conductor $720$
Sign $0.173 + 0.984i$
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  + (1.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−2 − 3.46i)7-s + (1.5 − 2.59i)9-s + (1.5 + 2.59i)11-s + (2 − 3.46i)13-s + 1.73i·15-s + 3·17-s − 5·19-s + (−6 − 3.46i)21-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s − 5.19i·27-s + (−3 − 5.19i)29-s + (1 − 1.73i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.223 + 0.387i)5-s + (−0.755 − 1.30i)7-s + (0.5 − 0.866i)9-s + (0.452 + 0.783i)11-s + (0.554 − 0.960i)13-s + 0.447i·15-s + 0.727·17-s − 1.14·19-s + (−1.30 − 0.755i)21-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s − 0.999i·27-s + (−0.557 − 0.964i)29-s + (0.179 − 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37216 - 1.15138i\)
\(L(\frac12)\) \(\approx\) \(1.37216 - 1.15138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (2 + 3.46i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.5 - 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (5.5 + 9.52i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.21289780206880567216753569045, −9.417541308764779902415699585209, −8.344613829901136435493235014713, −7.55870232114401753705904273887, −6.88314253294859671971811849541, −6.12212245074065086345496674465, −4.28873226265988229012981394407, −3.63661034085897527631193317907, −2.54007572311196241797055661688, −0.886081282075011703012248408510, 1.83632829658934941938625660347, 3.16098389543368325827477851598, 3.87818085717095173237037694826, 5.17998351658346428593068851196, 6.08023312401024454537753387710, 7.17977942168347120137183831319, 8.435858603904231763429045115396, 8.948955554398129765124944093313, 9.353629583556061453203500068952, 10.49897421605245869236952277247

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