Properties

Label 2-720-20.3-c1-0-9
Degree $2$
Conductor $720$
Sign $-0.0898 + 0.995i$
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  + (−2 + i)5-s + (−5 − 5i)13-s + (5 − 5i)17-s + (3 − 4i)25-s − 4i·29-s + (5 − 5i)37-s − 8·41-s − 7i·49-s + (−5 − 5i)53-s − 12·61-s + (15 + 5i)65-s + (5 + 5i)73-s + (−5 + 15i)85-s + 16i·89-s + (5 − 5i)97-s + ⋯
L(s)  = 1  + (−0.894 + 0.447i)5-s + (−1.38 − 1.38i)13-s + (1.21 − 1.21i)17-s + (0.600 − 0.800i)25-s − 0.742i·29-s + (0.821 − 0.821i)37-s − 1.24·41-s i·49-s + (−0.686 − 0.686i)53-s − 1.53·61-s + (1.86 + 0.620i)65-s + (0.585 + 0.585i)73-s + (−0.542 + 1.62i)85-s + 1.69i·89-s + (0.507 − 0.507i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0898 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.0898 + 0.995i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.0898 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.558495 - 0.611121i\)
\(L(\frac12)\) \(\approx\) \(0.558495 - 0.611121i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (2 - i)T \)
good7 \( 1 + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (5 + 5i)T + 13iT^{2} \)
17 \( 1 + (-5 + 5i)T - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (-5 + 5i)T - 37iT^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 + (-5 + 5i)T - 97iT^{2} \)
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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

−10.13501475744345699642418941512, −9.541155496830400545204402493216, −8.147513325142824767473202160957, −7.65505867975109223610531894634, −6.90350327093729244211903579518, −5.56676128846362329260603883788, −4.75168108136968396186086502134, −3.44409810353912618084281220687, −2.63212306855336792251199475316, −0.43842981696271729148507804072, 1.56151718046754628350246714915, 3.17301110087739539798135539875, 4.28295712761428366738200999132, 5.01419410160804988396575696349, 6.28591592061112047681363820206, 7.31861043615295832735158602451, 7.968249838097597296014444644572, 8.921850869445639689069105511047, 9.715288814539549451072271140271, 10.64065210830210825438694943800

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