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Properties

Label 2-720-15.14-c2-0-21
Degree $2$
Conductor $720$
Sign $-0.842 + 0.539i$
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  + (0.229 − 4.99i)5-s − 8.73i·7-s − 10.4i·11-s + 5.38i·13-s + 26.2·17-s + 2.70·19-s − 33.2·23-s + (−24.8 − 2.28i)25-s − 17.4i·29-s − 48.3·31-s + (−43.6 − 2.00i)35-s + 66.2i·37-s − 14.7i·41-s − 28.4i·43-s + 35.9·47-s + ⋯
L(s)  = 1  + (0.0458 − 0.998i)5-s − 1.24i·7-s − 0.953i·11-s + 0.414i·13-s + 1.54·17-s + 0.142·19-s − 1.44·23-s + (−0.995 − 0.0915i)25-s − 0.602i·29-s − 1.56·31-s + (−1.24 − 0.0572i)35-s + 1.79i·37-s − 0.359i·41-s − 0.661i·43-s + 0.764·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 + 0.539i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.842 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.842 + 0.539i$
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: \((2,\ 720,\ (\ :1),\ -0.842 + 0.539i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.349139855\)
\(L(\frac12)\) \(\approx\) \(1.349139855\)
\(L(2)\) 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 \( 1 + (-0.229 + 4.99i)T \)
good7 \( 1 + 8.73iT - 49T^{2} \)
11 \( 1 + 10.4iT - 121T^{2} \)
13 \( 1 - 5.38iT - 169T^{2} \)
17 \( 1 - 26.2T + 289T^{2} \)
19 \( 1 - 2.70T + 361T^{2} \)
23 \( 1 + 33.2T + 529T^{2} \)
29 \( 1 + 17.4iT - 841T^{2} \)
31 \( 1 + 48.3T + 961T^{2} \)
37 \( 1 - 66.2iT - 1.36e3T^{2} \)
41 \( 1 + 14.7iT - 1.68e3T^{2} \)
43 \( 1 + 28.4iT - 1.84e3T^{2} \)
47 \( 1 - 35.9T + 2.20e3T^{2} \)
53 \( 1 - 42.2T + 2.80e3T^{2} \)
59 \( 1 + 55.9iT - 3.48e3T^{2} \)
61 \( 1 + 96.1T + 3.72e3T^{2} \)
67 \( 1 + 15.4iT - 4.48e3T^{2} \)
71 \( 1 + 13.5iT - 5.04e3T^{2} \)
73 \( 1 + 63.7iT - 5.32e3T^{2} \)
79 \( 1 + 94.5T + 6.24e3T^{2} \)
83 \( 1 - 19.4T + 6.88e3T^{2} \)
89 \( 1 + 118. iT - 7.92e3T^{2} \)
97 \( 1 - 100. iT - 9.40e3T^{2} \)
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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

−9.918831248200409166756845426911, −9.000807142049985870522989895856, −8.042488365131365908359540233303, −7.47121094034183205293075019506, −6.17529501451856531313937925755, −5.35107025224837554037339712840, −4.21026928095079221022445614426, −3.45900661445928541514594057888, −1.57997579684810745318675853365, −0.47344445501328804642609137115, 1.89208143120996826081844911597, 2.86381937397530079513659924042, 3.94755686852305707710969228853, 5.49926100248261742318899445620, 5.90849085905738720967197077539, 7.22712795755972166732466208401, 7.77394220690894412262773216118, 8.956219903761548108715251149511, 9.793385423675127051846125627866, 10.43597445369790580321917725552

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