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Properties

Label 2-720-9.7-c1-0-10
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 + (−0.5 + 0.866i)7-s + (1.5 − 2.59i)9-s + (3 − 5.19i)11-s + (−1 − 1.73i)13-s + (−1.5 − 0.866i)15-s + 4·19-s − 1.73i·21-s + (4.5 + 7.79i)23-s + (−0.499 + 0.866i)25-s + 5.19i·27-s + (−1.5 + 2.59i)29-s + (−2 − 3.46i)31-s + 10.3i·33-s + ⋯
L(s)  = 1  + (−0.866 + 0.499i)3-s + (0.223 + 0.387i)5-s + (−0.188 + 0.327i)7-s + (0.5 − 0.866i)9-s + (0.904 − 1.56i)11-s + (−0.277 − 0.480i)13-s + (−0.387 − 0.223i)15-s + 0.917·19-s − 0.377i·21-s + (0.938 + 1.62i)23-s + (−0.0999 + 0.173i)25-s + 0.999i·27-s + (−0.278 + 0.482i)29-s + (−0.359 − 0.622i)31-s + 1.80i·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

\(L(1)\) \(\approx\) \(1.20950 + 0.213268i\)
\(L(\frac12)\) \(\approx\) \(1.20950 + 0.213268i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-4.5 - 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
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

−10.62699894040910626305379731916, −9.465929525481541844971763147613, −9.170002270595554644066441672151, −7.75767531866467805113213700477, −6.75031775607795694985892920470, −5.80295349566630215202399716279, −5.37606225425339136159483735082, −3.86912304158204033681038489741, −3.05802772614047974712709437189, −1.00139764272227101364269385792, 1.05214766785694222593918158316, 2.30275679563339956255812454951, 4.22769782145272350158919093450, 4.85283731608582176135917784889, 5.99429893171094008643750801277, 6.97418829706202091809810403279, 7.34788386273809706028260722473, 8.723714346463893322233000053260, 9.677358948156739509564546648466, 10.27721825181985521898558990473

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