Properties

Degree $2$
Conductor $720$
Sign $0.173 - 0.984i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (1.5 + 2.59i)9-s + (−2.5 + 4.33i)11-s + 1.73i·15-s + 3·17-s − 5·19-s + (3 + 5.19i)23-s + (−0.499 + 0.866i)25-s + 5.19i·27-s + (5 − 8.66i)29-s + (−1 − 1.73i)31-s + (−7.5 + 4.33i)33-s + 4·37-s + (1.5 + 2.59i)41-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)3-s + (0.223 + 0.387i)5-s + (0.5 + 0.866i)9-s + (−0.753 + 1.30i)11-s + 0.447i·15-s + 0.727·17-s − 1.14·19-s + (0.625 + 1.08i)23-s + (−0.0999 + 0.173i)25-s + 0.999i·27-s + (0.928 − 1.60i)29-s + (−0.179 − 0.311i)31-s + (−1.30 + 0.753i)33-s + 0.657·37-s + (0.234 + 0.405i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.173 - 0.984i$
Motivic weight: \(1\)
Character: $\chi_{720} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52251 + 1.27753i\)
\(L(\frac12)\) \(\approx\) \(1.52251 + 1.27753i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5 + 8.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.5 + 2.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2 + 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14T + 71T^{2} \)
73 \( 1 + 15T + 73T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 + (-6.5 + 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.30167182590907834215945763363, −9.841654842723544589284375624556, −9.025303657793254598679494721108, −7.87465109184261690182734328560, −7.44372152204783509840973182660, −6.18736440487051839131936384388, −4.97085517556857291745178790127, −4.13254258241428061702997466158, −2.89651450367539122506627930403, −1.98991972099260763786963171994, 0.977723197946596028285680891564, 2.50226890156502896041659072926, 3.39268277321198310993092640098, 4.69898396479728600753142460449, 5.86297783752437163521518093388, 6.74371997742015913801222992217, 7.82352083783939644058943236720, 8.575926059178326469351972147858, 9.012308300768752647510164394594, 10.26020690497743039708356183530

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