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Properties

Label 2-720-9.4-c1-0-12
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 + (1.36 + 2.36i)7-s + (2.91 + 0.728i)9-s + (2.76 + 4.78i)11-s + (1.76 − 3.05i)13-s + (1.04 − 1.38i)15-s − 5.52·17-s − 7.52·19-s + (1.84 + 4.36i)21-s + (0.367 − 0.635i)23-s + (−0.499 − 0.866i)25-s + (4.84 + 1.86i)27-s + (2.23 + 3.86i)29-s + (3.76 − 6.51i)31-s + ⋯
L(s)  = 1  + (0.992 + 0.122i)3-s + (0.223 − 0.387i)5-s + (0.516 + 0.894i)7-s + (0.970 + 0.242i)9-s + (0.832 + 1.44i)11-s + (0.488 − 0.846i)13-s + (0.269 − 0.357i)15-s − 1.33·17-s − 1.72·19-s + (0.403 + 0.951i)21-s + (0.0765 − 0.132i)23-s + (−0.0999 − 0.173i)25-s + (0.933 + 0.359i)27-s + (0.414 + 0.718i)29-s + (0.675 − 1.17i)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

\(L(1)\) \(\approx\) \(2.31585 + 0.533438i\)
\(L(\frac12)\) \(\approx\) \(2.31585 + 0.533438i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.36 - 2.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.76 - 4.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.76 + 3.05i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.52T + 17T^{2} \)
19 \( 1 + 7.52T + 19T^{2} \)
23 \( 1 + (-0.367 + 0.635i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.23 - 3.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.76 + 6.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.05T + 37T^{2} \)
41 \( 1 + (-0.527 + 0.914i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.76 + 3.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.604 - 1.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.46T + 53T^{2} \)
59 \( 1 + (0.734 - 1.27i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.52 + 7.84i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.12 - 3.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.63 + 4.56i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (4.73 + 8.19i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.29166001296296105019879816852, −9.418541794119145513265930636731, −8.711624856967898952450214575953, −8.218777715459261943843116652164, −7.03980906589767019778850213102, −6.10827297395942878979235569582, −4.70866415139629774638641274673, −4.16478870422537977732953554380, −2.53488848349328676427986670165, −1.77910442680137950032208992551, 1.33320550377828390936248058356, 2.60612589317739140743151965330, 3.90044621568111851235717945358, 4.42804104836286265547457600029, 6.39454178173572746602462395319, 6.65042380225634144055671296517, 7.938819197855063873640143747854, 8.668218510191231148346061818420, 9.223318760686387412675244123240, 10.49420931517193138631159808366

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