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Properties

Label 2-720-5.3-c2-0-14
Degree $2$
Conductor $720$
Sign $0.0107 + 0.999i$
Analytic cond. $19.6185$
Root an. cond. $4.42928$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.22 + 2.67i)5-s + (−7.44 + 7.44i)7-s − 16.2·11-s + (12.2 + 12.2i)13-s + (7.55 − 7.55i)17-s − 14.4i·19-s + (−2.65 − 2.65i)23-s + (10.6 − 22.5i)25-s − 34.2i·29-s + 20.4·31-s + (11.5 − 51.3i)35-s + (7.34 − 7.34i)37-s − 25.5·41-s + (−25.1 − 25.1i)43-s + (22.0 − 22.0i)47-s + ⋯
L(s)  = 1  + (−0.844 + 0.534i)5-s + (−1.06 + 1.06i)7-s − 1.47·11-s + (0.942 + 0.942i)13-s + (0.444 − 0.444i)17-s − 0.762i·19-s + (−0.115 − 0.115i)23-s + (0.427 − 0.903i)25-s − 1.18i·29-s + 0.661·31-s + (0.330 − 1.46i)35-s + (0.198 − 0.198i)37-s − 0.622·41-s + (−0.583 − 0.583i)43-s + (0.469 − 0.469i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0107 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0107 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.0107 + 0.999i$
Analytic conductor: \(19.6185\)
Root analytic conductor: \(4.42928\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1),\ 0.0107 + 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4018380573\)
\(L(\frac12)\) \(\approx\) \(0.4018380573\)
\(L(2)\) 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 \( 1 \)
5 \( 1 + (4.22 - 2.67i)T \)
good7 \( 1 + (7.44 - 7.44i)T - 49iT^{2} \)
11 \( 1 + 16.2T + 121T^{2} \)
13 \( 1 + (-12.2 - 12.2i)T + 169iT^{2} \)
17 \( 1 + (-7.55 + 7.55i)T - 289iT^{2} \)
19 \( 1 + 14.4iT - 361T^{2} \)
23 \( 1 + (2.65 + 2.65i)T + 529iT^{2} \)
29 \( 1 + 34.2iT - 841T^{2} \)
31 \( 1 - 20.4T + 961T^{2} \)
37 \( 1 + (-7.34 + 7.34i)T - 1.36e3iT^{2} \)
41 \( 1 + 25.5T + 1.68e3T^{2} \)
43 \( 1 + (25.1 + 25.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-22.0 + 22.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (-35.3 - 35.3i)T + 2.80e3iT^{2} \)
59 \( 1 - 88.7iT - 3.48e3T^{2} \)
61 \( 1 + 102.T + 3.72e3T^{2} \)
67 \( 1 + (-24.6 + 24.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 77.9T + 5.04e3T^{2} \)
73 \( 1 + (44.1 + 44.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 48.4iT - 6.24e3T^{2} \)
83 \( 1 + (101. + 101. i)T + 6.88e3iT^{2} \)
89 \( 1 - 156. iT - 7.92e3T^{2} \)
97 \( 1 + (-55.4 + 55.4i)T - 9.40e3iT^{2} \)
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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

−10.04099392208203167059620257549, −9.076036603408236205545974934296, −8.314407973505644289460725713944, −7.35922489205873853187513133431, −6.46023760378483996186553556135, −5.63105816303375434854145779464, −4.39390382084973860312855012521, −3.18466394396777900951010880484, −2.46809476212701440704227604609, −0.16866056185410762920186167174, 1.03658453943481315404117392943, 3.14819061339686944959327875299, 3.73520681440827180616077063864, 4.95996692327316820208267255537, 5.94741295628558357013031174844, 7.06609803658186957308274947627, 7.954550289870036722796474919181, 8.406209953289711258984763092185, 9.776440200613835885480641549630, 10.43448203563002020001698859979

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