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Properties

Label 2-720-16.13-c1-0-18
Degree $2$
Conductor $720$
Sign $0.189 - 0.981i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.04 + 0.948i)2-s + (0.202 + 1.98i)4-s + (0.707 − 0.707i)5-s + 0.740i·7-s + (−1.67 + 2.27i)8-s + (1.41 − 0.0715i)10-s + (3.83 − 3.83i)11-s + (3.31 + 3.31i)13-s + (−0.701 + 0.776i)14-s + (−3.91 + 0.804i)16-s − 2.93·17-s + (5.02 + 5.02i)19-s + (1.54 + 1.26i)20-s + (7.65 − 0.388i)22-s + 5.45i·23-s + ⋯
L(s)  = 1  + (0.741 + 0.670i)2-s + (0.101 + 0.994i)4-s + (0.316 − 0.316i)5-s + 0.279i·7-s + (−0.592 + 0.805i)8-s + (0.446 − 0.0226i)10-s + (1.15 − 1.15i)11-s + (0.918 + 0.918i)13-s + (−0.187 + 0.207i)14-s + (−0.979 + 0.201i)16-s − 0.712·17-s + (1.15 + 1.15i)19-s + (0.346 + 0.282i)20-s + (1.63 − 0.0827i)22-s + 1.13i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.189 - 0.981i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92079 + 1.58622i\)
\(L(\frac12)\) \(\approx\) \(1.92079 + 1.58622i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.04 - 0.948i)T \)
3 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - 0.740iT - 7T^{2} \)
11 \( 1 + (-3.83 + 3.83i)T - 11iT^{2} \)
13 \( 1 + (-3.31 - 3.31i)T + 13iT^{2} \)
17 \( 1 + 2.93T + 17T^{2} \)
19 \( 1 + (-5.02 - 5.02i)T + 19iT^{2} \)
23 \( 1 - 5.45iT - 23T^{2} \)
29 \( 1 + (2.64 + 2.64i)T + 29iT^{2} \)
31 \( 1 + 5.94T + 31T^{2} \)
37 \( 1 + (0.479 - 0.479i)T - 37iT^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + (4.93 - 4.93i)T - 43iT^{2} \)
47 \( 1 - 8.15T + 47T^{2} \)
53 \( 1 + (-5.05 + 5.05i)T - 53iT^{2} \)
59 \( 1 + (3.83 - 3.83i)T - 59iT^{2} \)
61 \( 1 + (4.87 + 4.87i)T + 61iT^{2} \)
67 \( 1 + (3.99 + 3.99i)T + 67iT^{2} \)
71 \( 1 - 3.55iT - 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + (4.61 + 4.61i)T + 83iT^{2} \)
89 \( 1 + 2.62iT - 89T^{2} \)
97 \( 1 - 1.67T + 97T^{2} \)
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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

−10.87508434932444373192429055164, −9.213911364592723750108877079676, −9.001019830875147602063281078514, −7.927185358295777693173802167031, −6.89422574251617009507552659424, −5.97427706572527233520468306697, −5.50236241828604556907542449378, −4.04900901517196497971038674979, −3.46278290057292506707989862457, −1.71367045861928441117873627186, 1.20578843682141625512310803716, 2.52614435606623696040065778733, 3.66526223224764588759686447280, 4.58037431023084152085642403956, 5.60354919945044218610537867487, 6.62907474306967167674804988199, 7.24792284499818389348699007939, 8.869033105581739354277158187224, 9.503138324991173311201766778704, 10.46216297927657444970428514401

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