Properties

Label 2-720-5.3-c2-0-16
Degree $2$
Conductor $720$
Sign $0.437 + 0.899i$
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  + (−2.67 + 4.22i)5-s + (1.44 − 1.44i)7-s − 3.34·11-s + (−10.4 − 10.4i)13-s + (2.65 − 2.65i)17-s − 20.6i·19-s + (16.4 + 16.4i)23-s + (−10.6 − 22.5i)25-s + 0.853i·29-s + 18.6·31-s + (2.24 + 10i)35-s + (38.0 − 38.0i)37-s + 28.6·41-s + (−22.4 − 22.4i)43-s + (19.7 − 19.7i)47-s + ⋯
L(s)  = 1  + (−0.534 + 0.844i)5-s + (0.207 − 0.207i)7-s − 0.304·11-s + (−0.803 − 0.803i)13-s + (0.155 − 0.155i)17-s − 1.08i·19-s + (0.715 + 0.715i)23-s + (−0.427 − 0.903i)25-s + 0.0294i·29-s + 0.603·31-s + (0.0642 + 0.285i)35-s + (1.02 − 1.02i)37-s + 0.699·41-s + (−0.523 − 0.523i)43-s + (0.420 − 0.420i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.437 + 0.899i$
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: \((2,\ 720,\ (\ :1),\ 0.437 + 0.899i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.213634727\)
\(L(\frac12)\) \(\approx\) \(1.213634727\)
\(L(2)\) 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 \( 1 + (2.67 - 4.22i)T \)
good7 \( 1 + (-1.44 + 1.44i)T - 49iT^{2} \)
11 \( 1 + 3.34T + 121T^{2} \)
13 \( 1 + (10.4 + 10.4i)T + 169iT^{2} \)
17 \( 1 + (-2.65 + 2.65i)T - 289iT^{2} \)
19 \( 1 + 20.6iT - 361T^{2} \)
23 \( 1 + (-16.4 - 16.4i)T + 529iT^{2} \)
29 \( 1 - 0.853iT - 841T^{2} \)
31 \( 1 - 18.6T + 961T^{2} \)
37 \( 1 + (-38.0 + 38.0i)T - 1.36e3iT^{2} \)
41 \( 1 - 28.6T + 1.68e3T^{2} \)
43 \( 1 + (22.4 + 22.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-19.7 + 19.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (28.6 + 28.6i)T + 2.80e3iT^{2} \)
59 \( 1 + 111. iT - 3.48e3T^{2} \)
61 \( 1 - 94.0T + 3.72e3T^{2} \)
67 \( 1 + (-54.8 + 54.8i)T - 4.48e3iT^{2} \)
71 \( 1 + 68T + 5.04e3T^{2} \)
73 \( 1 + (39.7 + 39.7i)T + 5.32e3iT^{2} \)
79 \( 1 - 24.4iT - 6.24e3T^{2} \)
83 \( 1 + (21.1 + 21.1i)T + 6.88e3iT^{2} \)
89 \( 1 + 94.1iT - 7.92e3T^{2} \)
97 \( 1 + (-14.5 + 14.5i)T - 9.40e3iT^{2} \)
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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

−10.14043332104794517057953410221, −9.312308036473608598376546261658, −8.096305412038036137998147610046, −7.45999581863082301052571996383, −6.73476083580465543577626914816, −5.50691641841415878222794396133, −4.55092220807029211947299777248, −3.32118363187675438825751867826, −2.44634847105085855999083559655, −0.47763320420846956106601585202, 1.19256630674800865850113568032, 2.62672157095913647044793128373, 4.08312257004678563274434663325, 4.80518365547845531956275964306, 5.78836042259144113437244386964, 6.95939220796836041625517202984, 7.920439391418583364601615519319, 8.549916392304715827508919156869, 9.471366131798006427369951062329, 10.25651335612086679941787777433

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