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Properties

Label 2-720-240.179-c1-0-21
Degree $2$
Conductor $720$
Sign $0.734 + 0.678i$
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  + (−1.41 + 0.0371i)2-s + (1.99 − 0.104i)4-s + (−1.81 − 1.31i)5-s + 1.40i·7-s + (−2.81 + 0.222i)8-s + (2.60 + 1.78i)10-s + (0.176 − 0.176i)11-s + (−1.72 + 1.72i)13-s + (−0.0522 − 1.98i)14-s + (3.97 − 0.419i)16-s + 3.15·17-s + (3.47 − 3.47i)19-s + (−3.75 − 2.43i)20-s + (−0.242 + 0.255i)22-s + 1.97·23-s + ⋯
L(s)  = 1  + (−0.999 + 0.0262i)2-s + (0.998 − 0.0524i)4-s + (−0.809 − 0.587i)5-s + 0.531i·7-s + (−0.996 + 0.0786i)8-s + (0.824 + 0.565i)10-s + (0.0531 − 0.0531i)11-s + (−0.479 + 0.479i)13-s + (−0.0139 − 0.531i)14-s + (0.994 − 0.104i)16-s + 0.765·17-s + (0.796 − 0.796i)19-s + (−0.839 − 0.543i)20-s + (−0.0516 + 0.0544i)22-s + 0.412·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.734 + 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.724122 - 0.283045i\)
\(L(\frac12)\) \(\approx\) \(0.724122 - 0.283045i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0371i)T \)
3 \( 1 \)
5 \( 1 + (1.81 + 1.31i)T \)
good7 \( 1 - 1.40iT - 7T^{2} \)
11 \( 1 + (-0.176 + 0.176i)T - 11iT^{2} \)
13 \( 1 + (1.72 - 1.72i)T - 13iT^{2} \)
17 \( 1 - 3.15T + 17T^{2} \)
19 \( 1 + (-3.47 + 3.47i)T - 19iT^{2} \)
23 \( 1 - 1.97T + 23T^{2} \)
29 \( 1 + (-2.62 + 2.62i)T - 29iT^{2} \)
31 \( 1 + 5.95iT - 31T^{2} \)
37 \( 1 + (5.72 + 5.72i)T + 37iT^{2} \)
41 \( 1 - 0.159T + 41T^{2} \)
43 \( 1 + (-6.63 + 6.63i)T - 43iT^{2} \)
47 \( 1 - 1.15iT - 47T^{2} \)
53 \( 1 + (-5.35 + 5.35i)T - 53iT^{2} \)
59 \( 1 + (4.48 - 4.48i)T - 59iT^{2} \)
61 \( 1 + (-6.80 - 6.80i)T + 61iT^{2} \)
67 \( 1 + (9.97 + 9.97i)T + 67iT^{2} \)
71 \( 1 - 0.0951iT - 71T^{2} \)
73 \( 1 - 7.99T + 73T^{2} \)
79 \( 1 + 5.66iT - 79T^{2} \)
83 \( 1 + (-12.2 + 12.2i)T - 83iT^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−10.18184105121271680227213460111, −9.161164952932382803127837876989, −8.831423913094821573049792715713, −7.67441966947378870480757790498, −7.24278027817483963844130705596, −5.94938052129461384340784209306, −4.96855062984786396252410930387, −3.58396800214669864192040234517, −2.31129328282895300229121517991, −0.70366176528699210277382284450, 1.08003937269128432822542603991, 2.84793165407096880757168316377, 3.66323446802289591873103398279, 5.19737558890300217710690148647, 6.46806289064748476173006366357, 7.31841335630379075823947112818, 7.81887684191172852463756271118, 8.698870157303145056035081903824, 9.850917316826370819664668319481, 10.40715401444762303335712077289

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