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Properties

Label 2-720-9.7-c1-0-3
Degree $2$
Conductor $720$
Sign $0.600 - 0.799i$
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  + (−1.25 − 1.19i)3-s + (−0.5 − 0.866i)5-s + (−0.257 + 0.445i)7-s + (0.160 + 2.99i)9-s + (−1.66 + 2.87i)11-s + (0.660 + 1.14i)13-s + (−0.403 + 1.68i)15-s − 3.32·17-s + 1.32·19-s + (0.853 − 0.253i)21-s + (2.06 + 3.57i)23-s + (−0.499 + 0.866i)25-s + (3.36 − 3.95i)27-s + (0.693 − 1.20i)29-s + (4.36 + 7.56i)31-s + ⋯
L(s)  = 1  + (−0.725 − 0.687i)3-s + (−0.223 − 0.387i)5-s + (−0.0971 + 0.168i)7-s + (0.0534 + 0.998i)9-s + (−0.500 + 0.867i)11-s + (0.183 + 0.317i)13-s + (−0.104 + 0.434i)15-s − 0.805·17-s + 0.303·19-s + (0.186 − 0.0552i)21-s + (0.430 + 0.745i)23-s + (−0.0999 + 0.173i)25-s + (0.648 − 0.761i)27-s + (0.128 − 0.222i)29-s + (0.784 + 1.35i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.600 - 0.799i$
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: \((2,\ 720,\ (\ :1/2),\ 0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.722620 + 0.360802i\)
\(L(\frac12)\) \(\approx\) \(0.722620 + 0.360802i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 + 1.19i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (0.257 - 0.445i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.66 - 2.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.660 - 1.14i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.32T + 17T^{2} \)
19 \( 1 - 1.32T + 19T^{2} \)
23 \( 1 + (-2.06 - 3.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.693 + 1.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.36 - 7.56i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.292T + 37T^{2} \)
41 \( 1 + (-5.67 - 9.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.17 + 8.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.43 - 4.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.02T + 53T^{2} \)
59 \( 1 + (2.51 + 4.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.67 - 6.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.72 - 8.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.99T + 71T^{2} \)
73 \( 1 - 6.05T + 73T^{2} \)
79 \( 1 + (4.02 - 6.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.771 + 1.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-6.12 + 10.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.70525171303462648636572188555, −9.719409095317921043363323886424, −8.755792155573676840679164106874, −7.76830908152122681914388461073, −7.05823407064210228048257282495, −6.15183337783491834716995629004, −5.12726655516091498576272158387, −4.37232465587802153393268784546, −2.65971677587015032710465322033, −1.33665228382331609259776376972, 0.49820860618979817483233831968, 2.75750824543165746762763474346, 3.83624502821156579531255766328, 4.80260445143314679651618851916, 5.85704211914772515556949748622, 6.54172471271716298897241080198, 7.65895180935880655878283747214, 8.674786850980437026340809386775, 9.551500178335232241204847654909, 10.54142562058475938221215657856

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