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Properties

Label 2-720-9.4-c1-0-2
Degree $2$
Conductor $720$
Sign $0.514 - 0.857i$
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  + (−0.285 − 1.70i)3-s + (−0.5 + 0.866i)5-s + (0.714 + 1.23i)7-s + (−2.83 + 0.977i)9-s + (1.33 + 2.31i)11-s + (−2.33 + 4.04i)13-s + (1.62 + 0.606i)15-s + 2.67·17-s − 4.67·19-s + (1.90 − 1.57i)21-s + (−2.95 + 5.12i)23-s + (−0.499 − 0.866i)25-s + (2.48 + 4.56i)27-s + (4.74 + 8.21i)29-s + (3.48 − 6.02i)31-s + ⋯
L(s)  = 1  + (−0.165 − 0.986i)3-s + (−0.223 + 0.387i)5-s + (0.269 + 0.467i)7-s + (−0.945 + 0.325i)9-s + (0.402 + 0.697i)11-s + (−0.648 + 1.12i)13-s + (0.418 + 0.156i)15-s + 0.648·17-s − 1.07·19-s + (0.416 − 0.343i)21-s + (−0.616 + 1.06i)23-s + (−0.0999 − 0.173i)25-s + (0.477 + 0.878i)27-s + (0.881 + 1.52i)29-s + (0.625 − 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.514 - 0.857i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.514 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.932011 + 0.527397i\)
\(L(\frac12)\) \(\approx\) \(0.932011 + 0.527397i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.285 + 1.70i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-0.714 - 1.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.33 - 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.33 - 4.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.67T + 17T^{2} \)
19 \( 1 + 4.67T + 19T^{2} \)
23 \( 1 + (2.95 - 5.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.74 - 8.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.48 + 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.81T + 37T^{2} \)
41 \( 1 + (-0.735 + 1.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.235 - 0.408i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.47 - 6.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 + (0.571 - 0.990i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.26 - 2.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.29 - 5.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 1.71T + 73T^{2} \)
79 \( 1 + (0.143 + 0.249i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.14 + 3.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (3.91 + 6.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.68333011708339171020371445849, −9.628964937915036599448596301823, −8.728957323004139016903202571720, −7.80507307547934296426884176295, −7.03530755394904603980514786156, −6.34177032638197557858751049719, −5.26615346480369418599127647470, −4.10453125307918264367550344147, −2.60108512586986914105917959450, −1.63028642934465492738512578637, 0.57426046479232169103156144402, 2.74926327522566883746729215049, 3.91380197241202923089499091064, 4.68475032189276330535648479817, 5.64238742358479683346623850188, 6.57843248599704051303465688904, 8.077661509361867354848462173725, 8.405296302521596230321707248606, 9.563997279655505119297096559447, 10.38023186764362844957620501147

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