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Properties

Label 2-720-9.4-c1-0-10
Degree $2$
Conductor $720$
Sign $0.899 - 0.437i$
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  + (1.71 + 0.211i)3-s + (−0.5 + 0.866i)5-s + (0.719 + 1.24i)7-s + (2.91 + 0.728i)9-s + (−0.675 − 1.17i)11-s + (2.76 − 4.78i)13-s + (−1.04 + 1.38i)15-s + 4.82·17-s + 0.648·19-s + (0.972 + 2.29i)21-s + (−4.45 + 7.71i)23-s + (−0.499 − 0.866i)25-s + (4.84 + 1.86i)27-s + (3.58 + 6.21i)29-s + (−2.32 + 4.02i)31-s + ⋯
L(s)  = 1  + (0.992 + 0.122i)3-s + (−0.223 + 0.387i)5-s + (0.271 + 0.470i)7-s + (0.970 + 0.242i)9-s + (−0.203 − 0.353i)11-s + (0.766 − 1.32i)13-s + (−0.269 + 0.357i)15-s + 1.16·17-s + 0.148·19-s + (0.212 + 0.500i)21-s + (−0.928 + 1.60i)23-s + (−0.0999 − 0.173i)25-s + (0.933 + 0.359i)27-s + (0.665 + 1.15i)29-s + (−0.417 + 0.722i)31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.899 - 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16165 + 0.497920i\)
\(L(\frac12)\) \(\approx\) \(2.16165 + 0.497920i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.211i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-0.719 - 1.24i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.675 + 1.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.76 + 4.78i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.82T + 17T^{2} \)
19 \( 1 - 0.648T + 19T^{2} \)
23 \( 1 + (4.45 - 7.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.58 - 6.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.32 - 4.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.35T + 37T^{2} \)
41 \( 1 + (0.175 - 0.304i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.41 - 4.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.74 + 8.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.17T + 53T^{2} \)
59 \( 1 + (-0.734 + 1.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.34 + 5.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.21 + 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.22T + 71T^{2} \)
73 \( 1 + 4.34T + 73T^{2} \)
79 \( 1 + (6.52 + 11.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.63 + 4.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11T + 89T^{2} \)
97 \( 1 + (8.79 + 15.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.34159729681232736238992250833, −9.650847949646314291339375420757, −8.552503334199486491776243592446, −8.022458789454224536905103967188, −7.28664265947286117890951803377, −5.94081561031271891343627990941, −5.07306296214964577182422686612, −3.50136225628018852921702204756, −3.12838374884375120708902118041, −1.54876241146315335480021808419, 1.30019339387600981544516645947, 2.56219233112452255970624353897, 4.01044501010515525881183031923, 4.43605073535811177267110514698, 6.01941933423042035690636668866, 7.04475254805329521900538803160, 7.933164598151745661347333462346, 8.483897226997875623309169527282, 9.479286401036138155428046325983, 10.11228851927287940109618271134

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