Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5 $
Sign $0.419 - 0.907i$
Motivic weight 5
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−23.4 + 50.7i)5-s + 10.2i·7-s + 596.·11-s − 420. i·13-s + 974. i·17-s − 380.·19-s + 3.54e3i·23-s + (−2.02e3 − 2.37e3i)25-s + 5.44e3·29-s + 3.62e3·31-s + (−520. − 240. i)35-s − 1.75e3i·37-s − 263.·41-s − 1.44e4i·43-s − 2.34e4i·47-s + ⋯
L(s)  = 1  + (−0.419 + 0.907i)5-s + 0.0791i·7-s + 1.48·11-s − 0.690i·13-s + 0.817i·17-s − 0.241·19-s + 1.39i·23-s + (−0.648 − 0.760i)25-s + 1.20·29-s + 0.677·31-s + (−0.0718 − 0.0331i)35-s − 0.210i·37-s − 0.0245·41-s − 1.18i·43-s − 1.54i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.419 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
\( \varepsilon \)  =  $0.419 - 0.907i$
motivic weight  =  \(5\)
character  :  $\chi_{720} (289, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 720,\ (\ :5/2),\ 0.419 - 0.907i)\)
\(L(3)\)  \(\approx\)  \(2.180456982\)
\(L(\frac12)\)  \(\approx\)  \(2.180456982\)
\(L(\frac{7}{2})\)   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 \( 1 + (23.4 - 50.7i)T \)
good7 \( 1 - 10.2iT - 1.68e4T^{2} \)
11 \( 1 - 596.T + 1.61e5T^{2} \)
13 \( 1 + 420. iT - 3.71e5T^{2} \)
17 \( 1 - 974. iT - 1.41e6T^{2} \)
19 \( 1 + 380.T + 2.47e6T^{2} \)
23 \( 1 - 3.54e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.44e3T + 2.05e7T^{2} \)
31 \( 1 - 3.62e3T + 2.86e7T^{2} \)
37 \( 1 + 1.75e3iT - 6.93e7T^{2} \)
41 \( 1 + 263.T + 1.15e8T^{2} \)
43 \( 1 + 1.44e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.34e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.34e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.90e3T + 7.14e8T^{2} \)
61 \( 1 - 2.94e4T + 8.44e8T^{2} \)
67 \( 1 - 7.16e3iT - 1.35e9T^{2} \)
71 \( 1 + 8.13e4T + 1.80e9T^{2} \)
73 \( 1 - 5.51e4iT - 2.07e9T^{2} \)
79 \( 1 - 1.64e4T + 3.07e9T^{2} \)
83 \( 1 - 1.16e5iT - 3.93e9T^{2} \)
89 \( 1 - 9.93e4T + 5.58e9T^{2} \)
97 \( 1 + 6.29e4iT - 8.58e9T^{2} \)
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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

−9.993287014828751936402849190835, −8.861173133454724095286647330851, −8.098679467866470437476765828375, −7.05866697624885833369668296762, −6.44354446838590481680849960185, −5.44809373050384889549322766315, −4.01860627971141916130710081741, −3.45599059610337497261531802066, −2.17835924943414403065866381828, −0.888419825970146979649165215715, 0.60357518105335594277876384796, 1.45879027932858330539191127212, 2.90515951120082088713759945014, 4.36288203381711057270607237504, 4.53688005425024073516702846052, 6.06472744214054026746283679133, 6.79839216505260951577702818652, 7.84630503731692950983129998300, 8.845130782792315103017036181327, 9.208590023973272690019510471023

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