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Properties

Label 2-720-9.7-c1-0-11
Degree $2$
Conductor $720$
Sign $0.939 - 0.342i$
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  + (1.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−1.5 + 2.59i)7-s + (1.5 − 2.59i)9-s + (−1 + 1.73i)11-s + (1 + 1.73i)13-s + (1.5 + 0.866i)15-s + 4·17-s + 8·19-s + 5.19i·21-s + (1.5 + 2.59i)23-s + (−0.499 + 0.866i)25-s − 5.19i·27-s + (0.5 − 0.866i)29-s + 3.46i·33-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.223 + 0.387i)5-s + (−0.566 + 0.981i)7-s + (0.5 − 0.866i)9-s + (−0.301 + 0.522i)11-s + (0.277 + 0.480i)13-s + (0.387 + 0.223i)15-s + 0.970·17-s + 1.83·19-s + 1.13i·21-s + (0.312 + 0.541i)23-s + (−0.0999 + 0.173i)25-s − 0.999i·27-s + (0.0928 − 0.160i)29-s + 0.603i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04157 + 0.359985i\)
\(L(\frac12)\) \(\approx\) \(2.04157 + 0.359985i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + (3 - 5.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.09150939549746597927053920240, −9.547609695683098551353532841257, −8.849943478974440186982724255510, −7.77650838471059596261502091712, −7.10679583974709831986743781916, −6.12156519138280435750732477786, −5.17055850751330882699124209379, −3.51759090327571145232493760843, −2.83625030546701733890904303266, −1.60763789755408143790910168679, 1.13874338536850994095405650931, 2.99728129544353360426983778020, 3.59621252759044902344282648574, 4.83168194940643303100738536912, 5.73892550641310864587451372998, 7.12820048389790173798542229356, 7.82029979437575278300668026496, 8.695231987502861283142760352908, 9.608617443960571825905924154476, 10.18892888898760967169349237883

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