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Properties

Label 2-720-5.4-c3-0-32
Degree $2$
Conductor $720$
Sign $0.178 + 0.983i$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 11i)5-s + 2i·7-s + 70·11-s − 54i·13-s + 22i·17-s + 24·19-s + 100i·23-s + (−117 + 44i)25-s + 216·29-s − 208·31-s + (22 − 4i)35-s − 254i·37-s + 206·41-s + 292i·43-s − 320i·47-s + ⋯
L(s)  = 1  + (−0.178 − 0.983i)5-s + 0.107i·7-s + 1.91·11-s − 1.15i·13-s + 0.313i·17-s + 0.289·19-s + 0.906i·23-s + (−0.936 + 0.351i)25-s + 1.38·29-s − 1.20·31-s + (0.106 − 0.0193i)35-s − 1.12i·37-s + 0.784·41-s + 1.03i·43-s − 0.993i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.178 + 0.983i$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ 0.178 + 0.983i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.075168328\)
\(L(\frac12)\) \(\approx\) \(2.075168328\)
\(L(\frac{5}{2})\) 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 \( 1 + (2 + 11i)T \)
good7 \( 1 - 2iT - 343T^{2} \)
11 \( 1 - 70T + 1.33e3T^{2} \)
13 \( 1 + 54iT - 2.19e3T^{2} \)
17 \( 1 - 22iT - 4.91e3T^{2} \)
19 \( 1 - 24T + 6.85e3T^{2} \)
23 \( 1 - 100iT - 1.21e4T^{2} \)
29 \( 1 - 216T + 2.43e4T^{2} \)
31 \( 1 + 208T + 2.97e4T^{2} \)
37 \( 1 + 254iT - 5.06e4T^{2} \)
41 \( 1 - 206T + 6.89e4T^{2} \)
43 \( 1 - 292iT - 7.95e4T^{2} \)
47 \( 1 + 320iT - 1.03e5T^{2} \)
53 \( 1 + 402iT - 1.48e5T^{2} \)
59 \( 1 - 370T + 2.05e5T^{2} \)
61 \( 1 + 550T + 2.26e5T^{2} \)
67 \( 1 + 728iT - 3.00e5T^{2} \)
71 \( 1 + 540T + 3.57e5T^{2} \)
73 \( 1 + 604iT - 3.89e5T^{2} \)
79 \( 1 - 792T + 4.93e5T^{2} \)
83 \( 1 + 404iT - 5.71e5T^{2} \)
89 \( 1 + 938T + 7.04e5T^{2} \)
97 \( 1 - 56iT - 9.12e5T^{2} \)
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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

−9.568100281711732189548584610546, −9.048788365532477163244870384900, −8.200948299399744884715549749707, −7.28970608964527741928219389677, −6.15067808331699079056410598138, −5.34157028449350730733491070418, −4.23169931365293033787417860015, −3.39744716806499982102542324336, −1.66833690159052246119388559745, −0.66583347662454195314243970244, 1.20951081433644063263851661442, 2.54958695918014148132565777785, 3.77381393481421268474641196108, 4.46593762602424361006896742301, 6.06182631274567230773366555169, 6.73957078026412521026314925313, 7.31706722832136606978202817879, 8.653004967483442220759748084364, 9.316450769193020078576240448388, 10.21494300752468365668362930674

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