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Properties

Label 2-720-60.59-c1-0-4
Degree $2$
Conductor $720$
Sign $0.805 - 0.592i$
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  + (2.12 + 0.707i)5-s + 6i·13-s + 4.24·17-s + (3.99 + 3i)25-s − 9.89i·29-s + 12i·37-s + 1.41i·41-s − 7·49-s + 12.7·53-s + 10·61-s + (−4.24 + 12.7i)65-s − 6i·73-s + (8.99 + 3i)85-s − 18.3i·89-s − 18i·97-s + ⋯
L(s)  = 1  + (0.948 + 0.316i)5-s + 1.66i·13-s + 1.02·17-s + (0.799 + 0.600i)25-s − 1.83i·29-s + 1.97i·37-s + 0.220i·41-s − 49-s + 1.74·53-s + 1.28·61-s + (−0.526 + 1.57i)65-s − 0.702i·73-s + (0.976 + 0.325i)85-s − 1.94i·89-s − 1.82i·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.805 - 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71475 + 0.562135i\)
\(L(\frac12)\) \(\approx\) \(1.71475 + 0.562135i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-2.12 - 0.707i)T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 9.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 12iT - 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 12.7T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 18.3iT - 89T^{2} \)
97 \( 1 + 18iT - 97T^{2} \)
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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

−10.22858873523728790655361789079, −9.768598938373104752985351775841, −8.939184574592813099381626688038, −7.903155010760251119685884814232, −6.79930325742883355687773580678, −6.18471188475883276033132270927, −5.14070764084516448705072179779, −4.03451212860242714545431697113, −2.69317707565741143120947844776, −1.55447840584404894572880174943, 1.07426458548525875510480035162, 2.55753157320541996864156812572, 3.67271042205855361107822824269, 5.33153412986997489846865611157, 5.50458999491880117101512860504, 6.78427968625620666457265771955, 7.78783864327914372609048335024, 8.658597468922702073777967107106, 9.531482242230564625049988797885, 10.34256590522793512226580418055

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