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Properties

Label 2-720-80.67-c1-0-15
Degree $2$
Conductor $720$
Sign $-0.104 - 0.994i$
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  + (−0.687 + 1.23i)2-s + (−1.05 − 1.69i)4-s + (2.07 + 0.832i)5-s + (2.83 + 2.83i)7-s + (2.82 − 0.134i)8-s + (−2.45 + 1.99i)10-s + (−1.95 − 1.95i)11-s − 2.05·13-s + (−5.45 + 1.55i)14-s + (−1.77 + 3.58i)16-s + (4.06 + 4.06i)17-s + (0.683 + 0.683i)19-s + (−0.773 − 4.40i)20-s + (3.76 − 1.07i)22-s + (4.95 − 4.95i)23-s + ⋯
L(s)  = 1  + (−0.486 + 0.873i)2-s + (−0.527 − 0.849i)4-s + (0.928 + 0.372i)5-s + (1.07 + 1.07i)7-s + (0.998 − 0.0473i)8-s + (−0.776 + 0.630i)10-s + (−0.590 − 0.590i)11-s − 0.569·13-s + (−1.45 + 0.415i)14-s + (−0.444 + 0.895i)16-s + (0.986 + 0.986i)17-s + (0.156 + 0.156i)19-s + (−0.173 − 0.984i)20-s + (0.803 − 0.228i)22-s + (1.03 − 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.104 - 0.994i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.104 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.970798 + 1.07807i\)
\(L(\frac12)\) \(\approx\) \(0.970798 + 1.07807i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.687 - 1.23i)T \)
3 \( 1 \)
5 \( 1 + (-2.07 - 0.832i)T \)
good7 \( 1 + (-2.83 - 2.83i)T + 7iT^{2} \)
11 \( 1 + (1.95 + 1.95i)T + 11iT^{2} \)
13 \( 1 + 2.05T + 13T^{2} \)
17 \( 1 + (-4.06 - 4.06i)T + 17iT^{2} \)
19 \( 1 + (-0.683 - 0.683i)T + 19iT^{2} \)
23 \( 1 + (-4.95 + 4.95i)T - 23iT^{2} \)
29 \( 1 + (-0.835 + 0.835i)T - 29iT^{2} \)
31 \( 1 + 2.35iT - 31T^{2} \)
37 \( 1 + 4.54T + 37T^{2} \)
41 \( 1 - 5.07iT - 41T^{2} \)
43 \( 1 + 0.849T + 43T^{2} \)
47 \( 1 + (2.72 - 2.72i)T - 47iT^{2} \)
53 \( 1 - 5.17iT - 53T^{2} \)
59 \( 1 + (4.16 - 4.16i)T - 59iT^{2} \)
61 \( 1 + (-5.55 - 5.55i)T + 61iT^{2} \)
67 \( 1 + 1.73T + 67T^{2} \)
71 \( 1 + 2.33T + 71T^{2} \)
73 \( 1 + (-4.39 - 4.39i)T + 73iT^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 2.75iT - 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + (3.52 + 3.52i)T + 97iT^{2} \)
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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

−10.45940158403370817596792358674, −9.688771796564745847562169286542, −8.705835902445735545975709077741, −8.200105527957396334188548855279, −7.20796416437354057651093120976, −6.04771336952392274650567797743, −5.53936843816779923648175174278, −4.72093514543868303199335841488, −2.76054487415168912521005672894, −1.52482121493592878790522286555, 1.02345000969431975755373130392, 2.07568815929337962910680803754, 3.37838971813390912269727949446, 4.88928125048541375206103313197, 5.10851621330088918738541435901, 7.12189763557809650484809796729, 7.60289395812264194587566295817, 8.637360843290244830403494143879, 9.602850378313199979872679048232, 10.13179390974259635540469306083

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