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Properties

Label 2-720-5.2-c2-0-1
Degree $2$
Conductor $720$
Sign $-0.981 + 0.189i$
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  + (−1.77 + 4.67i)5-s + (−2.55 − 2.55i)7-s + 8.24·11-s + (−12.2 + 12.2i)13-s + (12.4 + 12.4i)17-s − 34.4i·19-s + (−17.3 + 17.3i)23-s + (−18.6 − 16.5i)25-s + 9.75i·29-s − 28.4·31-s + (16.4 − 7.39i)35-s + (−7.34 − 7.34i)37-s − 74.4·41-s + (−34.8 + 34.8i)43-s + (−22.0 − 22.0i)47-s + ⋯
L(s)  = 1  + (−0.355 + 0.934i)5-s + (−0.364 − 0.364i)7-s + 0.749·11-s + (−0.942 + 0.942i)13-s + (0.732 + 0.732i)17-s − 1.81i·19-s + (−0.754 + 0.754i)23-s + (−0.747 − 0.663i)25-s + 0.336i·29-s − 0.919·31-s + (0.469 − 0.211i)35-s + (−0.198 − 0.198i)37-s − 1.81·41-s + (−0.811 + 0.811i)43-s + (−0.469 − 0.469i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.981 + 0.189i$
Analytic conductor: \(19.6185\)
Root analytic conductor: \(4.42928\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1),\ -0.981 + 0.189i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2188230152\)
\(L(\frac12)\) \(\approx\) \(0.2188230152\)
\(L(2)\) 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 \( 1 + (1.77 - 4.67i)T \)
good7 \( 1 + (2.55 + 2.55i)T + 49iT^{2} \)
11 \( 1 - 8.24T + 121T^{2} \)
13 \( 1 + (12.2 - 12.2i)T - 169iT^{2} \)
17 \( 1 + (-12.4 - 12.4i)T + 289iT^{2} \)
19 \( 1 + 34.4iT - 361T^{2} \)
23 \( 1 + (17.3 - 17.3i)T - 529iT^{2} \)
29 \( 1 - 9.75iT - 841T^{2} \)
31 \( 1 + 28.4T + 961T^{2} \)
37 \( 1 + (7.34 + 7.34i)T + 1.36e3iT^{2} \)
41 \( 1 + 74.4T + 1.68e3T^{2} \)
43 \( 1 + (34.8 - 34.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (22.0 + 22.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-64.6 + 64.6i)T - 2.80e3iT^{2} \)
59 \( 1 + 15.2iT - 3.48e3T^{2} \)
61 \( 1 + 53.5T + 3.72e3T^{2} \)
67 \( 1 + (4.69 + 4.69i)T + 4.48e3iT^{2} \)
71 \( 1 + 117.T + 5.04e3T^{2} \)
73 \( 1 + (-34.1 + 34.1i)T - 5.32e3iT^{2} \)
79 \( 1 + 0.494iT - 6.24e3T^{2} \)
83 \( 1 + (18.3 - 18.3i)T - 6.88e3iT^{2} \)
89 \( 1 - 136. iT - 7.92e3T^{2} \)
97 \( 1 + (-94.5 - 94.5i)T + 9.40e3iT^{2} \)
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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

−10.60583017264865264192242494536, −9.853111833336586078301783186345, −9.094008301710103372346950588687, −7.940148368482758375650542210903, −6.89038332023591850631178077748, −6.69638078390002058296986975824, −5.24712550738145441660593299032, −4.04501732731860679172675929202, −3.23784908268918521262021775181, −1.88727484598936372705706258436, 0.07470361966203165666392953380, 1.58093872976349298349981526199, 3.14460744902781885351318035297, 4.17814715460686224811432416465, 5.28924878489053580284845005334, 5.98182438343354281300915491583, 7.30257717053389562407704792595, 8.066307608270436158183899337056, 8.865073924024860694943180658420, 9.802130983707324668416952531063

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