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Properties

Label 2-720-20.19-c2-0-15
Degree $2$
Conductor $720$
Sign $0.948 - 0.316i$
Analytic cond. $19.6185$
Root an. cond. $4.42928$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 4.89i)5-s + 8.48·7-s + 13.8i·11-s + 9.79i·13-s + 19.5i·17-s + 13.8i·19-s + 25.4·23-s + (−22.9 − 9.79i)25-s + 22·29-s − 55.4i·31-s + (8.48 − 41.5i)35-s + 48.9i·37-s − 22·41-s + 59.3·43-s − 8.48·47-s + ⋯
L(s)  = 1  + (0.200 − 0.979i)5-s + 1.21·7-s + 1.25i·11-s + 0.753i·13-s + 1.15i·17-s + 0.729i·19-s + 1.10·23-s + (−0.919 − 0.391i)25-s + 0.758·29-s − 1.78i·31-s + (0.242 − 1.18i)35-s + 1.32i·37-s − 0.536·41-s + 1.38·43-s − 0.180·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.948 - 0.316i$
Analytic conductor: \(19.6185\)
Root analytic conductor: \(4.42928\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1),\ 0.948 - 0.316i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.199100084\)
\(L(\frac12)\) \(\approx\) \(2.199100084\)
\(L(2)\) 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 \( 1 \)
5 \( 1 + (-1 + 4.89i)T \)
good7 \( 1 - 8.48T + 49T^{2} \)
11 \( 1 - 13.8iT - 121T^{2} \)
13 \( 1 - 9.79iT - 169T^{2} \)
17 \( 1 - 19.5iT - 289T^{2} \)
19 \( 1 - 13.8iT - 361T^{2} \)
23 \( 1 - 25.4T + 529T^{2} \)
29 \( 1 - 22T + 841T^{2} \)
31 \( 1 + 55.4iT - 961T^{2} \)
37 \( 1 - 48.9iT - 1.36e3T^{2} \)
41 \( 1 + 22T + 1.68e3T^{2} \)
43 \( 1 - 59.3T + 1.84e3T^{2} \)
47 \( 1 + 8.48T + 2.20e3T^{2} \)
53 \( 1 + 29.3iT - 2.80e3T^{2} \)
59 \( 1 - 13.8iT - 3.48e3T^{2} \)
61 \( 1 - 46T + 3.72e3T^{2} \)
67 \( 1 - 59.3T + 4.48e3T^{2} \)
71 \( 1 + 27.7iT - 5.04e3T^{2} \)
73 \( 1 - 78.3iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 - 76.3T + 6.88e3T^{2} \)
89 \( 1 + 146T + 7.92e3T^{2} \)
97 \( 1 + 58.7iT - 9.40e3T^{2} \)
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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

−10.12314444933941136286159186327, −9.408567079873112220094318648353, −8.417086141303152316892211975458, −7.896748022563885638265485421799, −6.76587457971949025501721229378, −5.61012994713665419596218429139, −4.67943389130162563053487187332, −4.12173243985739374354569372247, −2.12328433888116365851164971099, −1.30989940018502029144559718886, 0.898934350621925357743928838947, 2.54929407515441619522061153894, 3.36009374717947888408977617960, 4.86907727573428887911347332534, 5.59047393734977297073220558731, 6.75674326683683776189357233670, 7.50687693131414763856887119901, 8.439404926580497552975778004389, 9.213650667793910855640697119645, 10.46104089603655875300388917027

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