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Properties

Label 2-720-9.7-c1-0-23
Degree $2$
Conductor $720$
Sign $-0.407 + 0.913i$
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  + (1.04 − 1.38i)3-s + (−0.5 − 0.866i)5-s + (2.04 − 3.53i)7-s + (−0.824 − 2.88i)9-s + (−0.675 + 1.17i)11-s + (−0.324 − 0.561i)13-s + (−1.71 − 0.211i)15-s − 1.35·17-s − 0.648·19-s + (−2.76 − 6.51i)21-s + (2.39 + 4.14i)23-s + (−0.499 + 0.866i)25-s + (−4.84 − 1.86i)27-s + (−1.93 + 3.35i)29-s + (−3.84 − 6.66i)31-s + ⋯
L(s)  = 1  + (0.602 − 0.798i)3-s + (−0.223 − 0.387i)5-s + (0.772 − 1.33i)7-s + (−0.274 − 0.961i)9-s + (−0.203 + 0.353i)11-s + (−0.0898 − 0.155i)13-s + (−0.443 − 0.0547i)15-s − 0.327·17-s − 0.148·19-s + (−0.602 − 1.42i)21-s + (0.499 + 0.864i)23-s + (−0.0999 + 0.173i)25-s + (−0.933 − 0.359i)27-s + (−0.359 + 0.623i)29-s + (−0.691 − 1.19i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.407 + 0.913i$
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: \((2,\ 720,\ (\ :1/2),\ -0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951032 - 1.46606i\)
\(L(\frac12)\) \(\approx\) \(0.951032 - 1.46606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 + 1.38i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-2.04 + 3.53i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.675 - 1.17i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.324 + 0.561i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.35T + 17T^{2} \)
19 \( 1 + 0.648T + 19T^{2} \)
23 \( 1 + (-2.39 - 4.14i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.93 - 3.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.84 + 6.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.52T + 37T^{2} \)
41 \( 1 + (-0.0898 - 0.155i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.410 - 0.710i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.45 + 9.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.17T + 53T^{2} \)
59 \( 1 + (-2.08 - 3.61i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.91 + 3.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.07 - 7.05i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.11T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 + (-5.17 + 8.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.12 - 10.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-6.79 + 11.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.08902628201452146179412503574, −9.144048152059281040570010868296, −8.242497040577506439988768836385, −7.46635563133577485983659444150, −7.06347550424454083053148122470, −5.67318925953923448678896681943, −4.46495593977468572842592703825, −3.59838000700437887678937448938, −2.08276780125918305618767738282, −0.870651162482362609620855669491, 2.17134356011767976614715632775, 2.99108374974661474752050282797, 4.28202881620326756918688290138, 5.15833253412465989495610734665, 6.07772386086957456074332532470, 7.42651232537103920694377024775, 8.378152597190558272901512080647, 8.849963587029331797877958918910, 9.708829565411480206519710973581, 10.81562342166401617149146377690

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