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Properties

Label 2-720-20.7-c1-0-12
Degree $2$
Conductor $720$
Sign $-0.525 + 0.850i$
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 − 2i)5-s + (1 − i)13-s + (−5 − 5i)17-s + (−3 + 4i)25-s − 10i·29-s + (−7 − 7i)37-s − 10·41-s + 7i·49-s + (5 − 5i)53-s + 12·61-s + (−3 − i)65-s + (11 − 11i)73-s + (−5 + 15i)85-s + 10i·89-s + (−13 − 13i)97-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s + (0.277 − 0.277i)13-s + (−1.21 − 1.21i)17-s + (−0.600 + 0.800i)25-s − 1.85i·29-s + (−1.15 − 1.15i)37-s − 1.56·41-s + i·49-s + (0.686 − 0.686i)53-s + 1.53·61-s + (−0.372 − 0.124i)65-s + (1.28 − 1.28i)73-s + (−0.542 + 1.62i)85-s + 1.05i·89-s + (−1.31 − 1.31i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.525 + 0.850i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.458065 - 0.821587i\)
\(L(\frac12)\) \(\approx\) \(0.458065 - 0.821587i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (1 + 2i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (5 + 5i)T + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + 10iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + (7 + 7i)T + 37iT^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-11 + 11i)T - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + (13 + 13i)T + 97iT^{2} \)
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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

−10.01494075156774709424674860623, −9.155922408831269568252927866396, −8.480562004187249511904055025426, −7.60198276477324176105330257627, −6.65049688413170539207160463909, −5.46976334895499297820517657150, −4.63988777221551882064743399918, −3.67102638766190293415429729947, −2.18720762862321371885776946629, −0.47540063476749427317373765702, 1.86923672087254418009846353794, 3.23630675826133321837138035102, 4.09557281354080656170450891442, 5.31758959749386313518572698943, 6.63553547403116105017187727342, 6.91800161053221953059438822094, 8.256940296972529450224820556715, 8.780335037974028390004830251118, 10.08844439675758383541538758720, 10.69070048906599955411700646339

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