Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5 $
Sign $-0.235 - 0.971i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.1 + 54.3i)5-s − 146. i·7-s + 191.·11-s − 83.9i·13-s + 2.00e3i·17-s + 677.·19-s − 1.29e3i·23-s + (−2.77e3 + 1.42e3i)25-s − 3.26e3·29-s − 6.15e3·31-s + (7.97e3 − 1.93e3i)35-s − 1.13e4i·37-s + 1.05e4·41-s + 1.29e4i·43-s + 9.52e3i·47-s + ⋯
L(s)  = 1  + (0.235 + 0.971i)5-s − 1.13i·7-s + 0.476·11-s − 0.137i·13-s + 1.67i·17-s + 0.430·19-s − 0.510i·23-s + (−0.889 + 0.457i)25-s − 0.721·29-s − 1.15·31-s + (1.10 − 0.266i)35-s − 1.36i·37-s + 0.984·41-s + 1.06i·43-s + 0.628i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $-0.235 - 0.971i$
motivic weight  =  \(5\)
character  :  $\chi_{720} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 720,\ (\ :5/2),\ -0.235 - 0.971i)\)
\(L(3)\)  \(\approx\)  \(1.646412068\)
\(L(\frac12)\)  \(\approx\)  \(1.646412068\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-13.1 - 54.3i)T \)
good7 \( 1 + 146. iT - 1.68e4T^{2} \)
11 \( 1 - 191.T + 1.61e5T^{2} \)
13 \( 1 + 83.9iT - 3.71e5T^{2} \)
17 \( 1 - 2.00e3iT - 1.41e6T^{2} \)
19 \( 1 - 677.T + 2.47e6T^{2} \)
23 \( 1 + 1.29e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.26e3T + 2.05e7T^{2} \)
31 \( 1 + 6.15e3T + 2.86e7T^{2} \)
37 \( 1 + 1.13e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.05e4T + 1.15e8T^{2} \)
43 \( 1 - 1.29e4iT - 1.47e8T^{2} \)
47 \( 1 - 9.52e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.47e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.82e4T + 7.14e8T^{2} \)
61 \( 1 + 3.58e3T + 8.44e8T^{2} \)
67 \( 1 + 2.17e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.13e4T + 1.80e9T^{2} \)
73 \( 1 - 1.33e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.59e4T + 3.07e9T^{2} \)
83 \( 1 - 5.33e4iT - 3.93e9T^{2} \)
89 \( 1 + 5.13e4T + 5.58e9T^{2} \)
97 \( 1 - 8.08e4iT - 8.58e9T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.01107623221740619884121549118, −9.189177067266470430787973864274, −7.952279362495052154422455909593, −7.27925718178122363516563610940, −6.44436110684618593924350566270, −5.63443166967463630863525127279, −4.10335060159638346953169000086, −3.59318526703577073335998981585, −2.23418420605270386679722770888, −1.08711662988951257461182819972, 0.36171891910173093535228290110, 1.58135566858768221600228768031, 2.61658104911642400989684470063, 3.89873082930688006727976039435, 5.18530626364323829444221298153, 5.49956990078726753464623906378, 6.74694169187807298907596136619, 7.74967700142122574007179509615, 8.829338826891346711026325831924, 9.234798881905663620289802308134

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