Properties

Label 2-720-15.14-c2-0-16
Degree $2$
Conductor $720$
Sign $-0.322 + 0.946i$
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  + (−4.79 + 1.41i)5-s + 6.78i·7-s + 9.89i·11-s − 20.3i·13-s − 19.1·17-s − 12·19-s + 9.59·23-s + (20.9 − 13.5i)25-s + 8.48i·29-s + 38·31-s + (−9.59 − 32.5i)35-s − 6.78i·37-s − 69.2i·41-s − 67.8i·43-s − 76.7·47-s + ⋯
L(s)  = 1  + (−0.959 + 0.282i)5-s + 0.968i·7-s + 0.899i·11-s − 1.56i·13-s − 1.12·17-s − 0.631·19-s + 0.417·23-s + (0.839 − 0.542i)25-s + 0.292i·29-s + 1.22·31-s + (−0.274 − 0.929i)35-s − 0.183i·37-s − 1.69i·41-s − 1.57i·43-s − 1.63·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.322 + 0.946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.322 + 0.946i$
Analytic conductor: \(19.6185\)
Root analytic conductor: \(4.42928\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1),\ -0.322 + 0.946i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5025837797\)
\(L(\frac12)\) \(\approx\) \(0.5025837797\)
\(L(2)\) 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 \( 1 + (4.79 - 1.41i)T \)
good7 \( 1 - 6.78iT - 49T^{2} \)
11 \( 1 - 9.89iT - 121T^{2} \)
13 \( 1 + 20.3iT - 169T^{2} \)
17 \( 1 + 19.1T + 289T^{2} \)
19 \( 1 + 12T + 361T^{2} \)
23 \( 1 - 9.59T + 529T^{2} \)
29 \( 1 - 8.48iT - 841T^{2} \)
31 \( 1 - 38T + 961T^{2} \)
37 \( 1 + 6.78iT - 1.36e3T^{2} \)
41 \( 1 + 69.2iT - 1.68e3T^{2} \)
43 \( 1 + 67.8iT - 1.84e3T^{2} \)
47 \( 1 + 76.7T + 2.20e3T^{2} \)
53 \( 1 + 2.80e3T^{2} \)
59 \( 1 + 83.4iT - 3.48e3T^{2} \)
61 \( 1 + 70T + 3.72e3T^{2} \)
67 \( 1 + 108. iT - 4.48e3T^{2} \)
71 \( 1 - 118. iT - 5.04e3T^{2} \)
73 \( 1 + 13.5iT - 5.32e3T^{2} \)
79 \( 1 + 30T + 6.24e3T^{2} \)
83 \( 1 + 134.T + 6.88e3T^{2} \)
89 \( 1 + 32.5iT - 7.92e3T^{2} \)
97 \( 1 + 94.9iT - 9.40e3T^{2} \)
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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

−10.08689064520638130126539339274, −8.874004923532646078489852204701, −8.308974044356005645880570108041, −7.36052330904968145794517324660, −6.51063338271153183142230852327, −5.35134387527944707559252729274, −4.45160594224813587226764565586, −3.23959928664010922161579251620, −2.22131690238259268019193647486, −0.18996158735494149256740326497, 1.24464943366062013008960699521, 2.98998293888909221110874308288, 4.28724527023839170645223133491, 4.54452813832720222889711051919, 6.31684737887153408583525080951, 6.90629747753969979542327658630, 7.967933201877765362433825704019, 8.648522487061147003466394797402, 9.518263618919365771785741319191, 10.66803749022949005545644049884

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