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Properties

Label 2-720-20.19-c2-0-28
Degree $2$
Conductor $720$
Sign $-0.399 + 0.916i$
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  + (2 − 4.58i)5-s + 3.46·7-s − 15.8i·11-s − 9.16i·13-s − 9.16i·17-s + 31.7i·19-s − 27.7·23-s + (−17 − 18.3i)25-s + 8·29-s + (6.92 − 15.8i)35-s − 45.8i·37-s − 50·41-s + 62.3·43-s + 48.4·47-s − 37·49-s + ⋯
L(s)  = 1  + (0.400 − 0.916i)5-s + 0.494·7-s − 1.44i·11-s − 0.705i·13-s − 0.539i·17-s + 1.67i·19-s − 1.20·23-s + (−0.680 − 0.733i)25-s + 0.275·29-s + (0.197 − 0.453i)35-s − 1.23i·37-s − 1.21·41-s + 1.45·43-s + 1.03·47-s − 0.755·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.399 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.399 + 0.916i$
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: \((2,\ 720,\ (\ :1),\ -0.399 + 0.916i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.671706077\)
\(L(\frac12)\) \(\approx\) \(1.671706077\)
\(L(2)\) 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 \( 1 + (-2 + 4.58i)T \)
good7 \( 1 - 3.46T + 49T^{2} \)
11 \( 1 + 15.8iT - 121T^{2} \)
13 \( 1 + 9.16iT - 169T^{2} \)
17 \( 1 + 9.16iT - 289T^{2} \)
19 \( 1 - 31.7iT - 361T^{2} \)
23 \( 1 + 27.7T + 529T^{2} \)
29 \( 1 - 8T + 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + 45.8iT - 1.36e3T^{2} \)
41 \( 1 + 50T + 1.68e3T^{2} \)
43 \( 1 - 62.3T + 1.84e3T^{2} \)
47 \( 1 - 48.4T + 2.20e3T^{2} \)
53 \( 1 + 27.4iT - 2.80e3T^{2} \)
59 \( 1 - 15.8iT - 3.48e3T^{2} \)
61 \( 1 + 26T + 3.72e3T^{2} \)
67 \( 1 + 55.4T + 4.48e3T^{2} \)
71 \( 1 - 95.2iT - 5.04e3T^{2} \)
73 \( 1 + 128. iT - 5.32e3T^{2} \)
79 \( 1 + 126. iT - 6.24e3T^{2} \)
83 \( 1 + 131.T + 6.88e3T^{2} \)
89 \( 1 - 86T + 7.92e3T^{2} \)
97 \( 1 + 109. iT - 9.40e3T^{2} \)
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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

−9.984983273489167289034549536783, −8.933195466043008725147376520206, −8.281605107383823189200707826579, −7.60443320646196760613427460405, −5.91701893899739685741103807724, −5.70167509585815542325247477659, −4.45390018446144366138502426902, −3.35559592860417335864960092190, −1.85136119243904226762972696511, −0.57535073069386502771676904811, 1.74411503909008007444689376533, 2.63900405355367211950264698925, 4.11122302901558982019052664272, 4.95540754247163696830167759984, 6.23713520733827214269704918690, 6.96463973678336517234389521577, 7.70302430309216305955090543030, 8.872181714388723811909978852215, 9.754738066679740224949796441472, 10.39790646529618695443765712504

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