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Properties

Label 2-720-80.29-c1-0-57
Degree $2$
Conductor $720$
Sign $-0.897 - 0.440i$
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  + (0.903 − 1.08i)2-s + (−0.368 − 1.96i)4-s + (−0.770 − 2.09i)5-s − 3.05·7-s + (−2.47 − 1.37i)8-s + (−2.98 − 1.05i)10-s + (1.80 − 1.80i)11-s + (−2.47 + 2.47i)13-s + (−2.75 + 3.31i)14-s + (−3.72 + 1.44i)16-s + 3.66i·17-s + (−2.31 − 2.31i)19-s + (−3.84 + 2.28i)20-s + (−0.334 − 3.60i)22-s + 4.86·23-s + ⋯
L(s)  = 1  + (0.638 − 0.769i)2-s + (−0.184 − 0.982i)4-s + (−0.344 − 0.938i)5-s − 1.15·7-s + (−0.873 − 0.486i)8-s + (−0.942 − 0.334i)10-s + (0.545 − 0.545i)11-s + (−0.685 + 0.685i)13-s + (−0.736 + 0.887i)14-s + (−0.932 + 0.361i)16-s + 0.889i·17-s + (−0.531 − 0.531i)19-s + (−0.859 + 0.511i)20-s + (−0.0713 − 0.768i)22-s + 1.01·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.897 - 0.440i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.897 - 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.219750 + 0.947081i\)
\(L(\frac12)\) \(\approx\) \(0.219750 + 0.947081i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.903 + 1.08i)T \)
3 \( 1 \)
5 \( 1 + (0.770 + 2.09i)T \)
good7 \( 1 + 3.05T + 7T^{2} \)
11 \( 1 + (-1.80 + 1.80i)T - 11iT^{2} \)
13 \( 1 + (2.47 - 2.47i)T - 13iT^{2} \)
17 \( 1 - 3.66iT - 17T^{2} \)
19 \( 1 + (2.31 + 2.31i)T + 19iT^{2} \)
23 \( 1 - 4.86T + 23T^{2} \)
29 \( 1 + (4.74 + 4.74i)T + 29iT^{2} \)
31 \( 1 - 1.86T + 31T^{2} \)
37 \( 1 + (5.40 + 5.40i)T + 37iT^{2} \)
41 \( 1 + 6.47iT - 41T^{2} \)
43 \( 1 + (-4.19 - 4.19i)T + 43iT^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 + (9.99 + 9.99i)T + 53iT^{2} \)
59 \( 1 + (-2.47 + 2.47i)T - 59iT^{2} \)
61 \( 1 + (-8.01 - 8.01i)T + 61iT^{2} \)
67 \( 1 + (-8.60 + 8.60i)T - 67iT^{2} \)
71 \( 1 + 6.63iT - 71T^{2} \)
73 \( 1 + 2.70T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + (-3.65 + 3.65i)T - 83iT^{2} \)
89 \( 1 - 13.1iT - 89T^{2} \)
97 \( 1 + 12.4iT - 97T^{2} \)
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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

−9.867068097701468822164160114929, −9.238588588754192350007260020596, −8.567484212946066745546580858216, −7.04064104704340323124969613529, −6.18788375063686908742575363627, −5.21605960638410594112235210926, −4.16429369750941733560135212114, −3.44842874653953661028394211639, −2.01124585142094760744527565077, −0.38911019641255047069984828694, 2.75117061201445462388866282955, 3.40847870178350871349141547531, 4.54423112780045855701798585512, 5.68560807885366588357771704433, 6.71800657967110285921331922323, 7.07398276896793417935277155298, 7.996530133840543992014022029159, 9.208991781010002770497439318531, 9.913688578844396288743277203300, 10.97412722671441759562174796801

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