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Properties

Label 2-720-9.7-c1-0-6
Degree $2$
Conductor $720$
Sign $0.939 - 0.342i$
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  + (−0.866 − 1.5i)3-s + (0.5 + 0.866i)5-s + (0.133 − 0.232i)7-s + (−1.5 + 2.59i)9-s + (−0.732 + 1.26i)11-s + (2.73 + 4.73i)13-s + (0.866 − 1.5i)15-s + 0.535·17-s + 2·19-s − 0.464·21-s + (1.86 + 3.23i)23-s + (−0.499 + 0.866i)25-s + 5.19·27-s + (−0.767 + 1.33i)29-s + (−1 − 1.73i)31-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (0.223 + 0.387i)5-s + (0.0506 − 0.0877i)7-s + (−0.5 + 0.866i)9-s + (−0.220 + 0.382i)11-s + (0.757 + 1.31i)13-s + (0.223 − 0.387i)15-s + 0.129·17-s + 0.458·19-s − 0.101·21-s + (0.389 + 0.673i)23-s + (−0.0999 + 0.173i)25-s + 1.00·27-s + (−0.142 + 0.246i)29-s + (−0.179 − 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25456 + 0.221213i\)
\(L(\frac12)\) \(\approx\) \(1.25456 + 0.221213i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-0.133 + 0.232i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.732 - 1.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.73 - 4.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.535T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + (-1.86 - 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.767 - 1.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + (4.96 + 8.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.26 - 3.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.133 + 0.232i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-7.19 - 12.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.23 + 7.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.13 - 5.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.46T + 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + (-7.73 + 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.59 - 11.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 + (-4.46 + 7.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.64678491763587008843098802905, −9.613256421081321748046623447947, −8.700922513128986030203641919452, −7.59991870314145153478055969415, −6.97179043446640638886547515686, −6.13113912867951698656711326412, −5.25155443306023009544516508255, −3.98767178447564564844249093783, −2.50941293676367291269323027473, −1.35318532116602086919910884241, 0.805421824636803346274993369744, 2.88065150988372629488412583131, 3.89745579472869260738067587030, 5.09109234301985860271388624928, 5.66276724249526166519175194963, 6.59447129674172733634572766983, 8.038074580991079178119216828698, 8.671602372428560625602676072133, 9.675256627148733638606678687370, 10.33561388236564536529115086111

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