Properties

Label 2-720-16.5-c1-0-38
Degree $2$
Conductor $720$
Sign $0.382 + 0.923i$
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  + 1.41·2-s + 2.00·4-s + (−0.707 − 0.707i)5-s − 4.82i·7-s + 2.82·8-s + (−1.00 − 1.00i)10-s + (−1.41 − 1.41i)11-s + (−0.585 + 0.585i)13-s − 6.82i·14-s + 4.00·16-s − 5.41·17-s + (3.82 − 3.82i)19-s + (−1.41 − 1.41i)20-s + (−2.00 − 2.00i)22-s + 5.41i·23-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.00·4-s + (−0.316 − 0.316i)5-s − 1.82i·7-s + 1.00·8-s + (−0.316 − 0.316i)10-s + (−0.426 − 0.426i)11-s + (−0.162 + 0.162i)13-s − 1.82i·14-s + 1.00·16-s − 1.31·17-s + (0.878 − 0.878i)19-s + (−0.316 − 0.316i)20-s + (−0.426 − 0.426i)22-s + 1.12i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16312 - 1.44535i\)
\(L(\frac12)\) \(\approx\) \(2.16312 - 1.44535i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + 4.82iT - 7T^{2} \)
11 \( 1 + (1.41 + 1.41i)T + 11iT^{2} \)
13 \( 1 + (0.585 - 0.585i)T - 13iT^{2} \)
17 \( 1 + 5.41T + 17T^{2} \)
19 \( 1 + (-3.82 + 3.82i)T - 19iT^{2} \)
23 \( 1 - 5.41iT - 23T^{2} \)
29 \( 1 + (-0.585 + 0.585i)T - 29iT^{2} \)
31 \( 1 - 3.65T + 31T^{2} \)
37 \( 1 + (-4.58 - 4.58i)T + 37iT^{2} \)
41 \( 1 + 4.82iT - 41T^{2} \)
43 \( 1 + (-3.65 - 3.65i)T + 43iT^{2} \)
47 \( 1 - 7.07T + 47T^{2} \)
53 \( 1 + (-4 - 4i)T + 53iT^{2} \)
59 \( 1 + (-7.41 - 7.41i)T + 59iT^{2} \)
61 \( 1 + (-9.48 + 9.48i)T - 61iT^{2} \)
67 \( 1 + (7.65 - 7.65i)T - 67iT^{2} \)
71 \( 1 - 8iT - 71T^{2} \)
73 \( 1 - 3.17iT - 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + (-3.07 + 3.07i)T - 83iT^{2} \)
89 \( 1 + 3.65iT - 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−10.54144164377034142358467421763, −9.600211087517009599667182154763, −8.254468957406241669483772980213, −7.29501512797475085966123544662, −6.89723847577198610426877296343, −5.59652841924609334790825675525, −4.52806857079287912577582144176, −3.97680140801205382686129819548, −2.78377945845621131275261528742, −1.02231951697759098449684668105, 2.21061334529709178231934295255, 2.82357140665363153117028514468, 4.20807900385427558275043333716, 5.20548752632221928563298398664, 5.96143437341199722836198636853, 6.82724929963524474710323231495, 7.896624193449233314849871874746, 8.737066591210459943392487870405, 9.842523997209935562367725258777, 10.81756783022908837450876434727

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