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Properties

Label 2-720-240.179-c1-0-26
Degree $2$
Conductor $720$
Sign $0.303 + 0.952i$
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  + (−1.22 − 0.698i)2-s + (1.02 + 1.71i)4-s + (−1.15 + 1.91i)5-s − 3.02i·7-s + (−0.0563 − 2.82i)8-s + (2.75 − 1.54i)10-s + (−1.30 + 1.30i)11-s + (1.56 − 1.56i)13-s + (−2.11 + 3.72i)14-s + (−1.90 + 3.51i)16-s + 2.98·17-s + (1.25 − 1.25i)19-s + (−4.47 − 0.0259i)20-s + (2.51 − 0.692i)22-s − 7.82·23-s + ⋯
L(s)  = 1  + (−0.869 − 0.494i)2-s + (0.511 + 0.859i)4-s + (−0.516 + 0.856i)5-s − 1.14i·7-s + (−0.0199 − 0.999i)8-s + (0.872 − 0.489i)10-s + (−0.393 + 0.393i)11-s + (0.434 − 0.434i)13-s + (−0.565 + 0.995i)14-s + (−0.476 + 0.878i)16-s + 0.724·17-s + (0.288 − 0.288i)19-s + (−0.999 − 0.00581i)20-s + (0.536 − 0.147i)22-s − 1.63·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.303 + 0.952i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.303 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.651353 - 0.476164i\)
\(L(\frac12)\) \(\approx\) \(0.651353 - 0.476164i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.698i)T \)
3 \( 1 \)
5 \( 1 + (1.15 - 1.91i)T \)
good7 \( 1 + 3.02iT - 7T^{2} \)
11 \( 1 + (1.30 - 1.30i)T - 11iT^{2} \)
13 \( 1 + (-1.56 + 1.56i)T - 13iT^{2} \)
17 \( 1 - 2.98T + 17T^{2} \)
19 \( 1 + (-1.25 + 1.25i)T - 19iT^{2} \)
23 \( 1 + 7.82T + 23T^{2} \)
29 \( 1 + (-7.12 + 7.12i)T - 29iT^{2} \)
31 \( 1 + 0.0502iT - 31T^{2} \)
37 \( 1 + (-6.22 - 6.22i)T + 37iT^{2} \)
41 \( 1 - 4.05T + 41T^{2} \)
43 \( 1 + (-6.18 + 6.18i)T - 43iT^{2} \)
47 \( 1 + 5.87iT - 47T^{2} \)
53 \( 1 + (-4.40 + 4.40i)T - 53iT^{2} \)
59 \( 1 + (-8.20 + 8.20i)T - 59iT^{2} \)
61 \( 1 + (4.93 + 4.93i)T + 61iT^{2} \)
67 \( 1 + (-1.29 - 1.29i)T + 67iT^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 + 3.85T + 73T^{2} \)
79 \( 1 - 1.07iT - 79T^{2} \)
83 \( 1 + (7.16 - 7.16i)T - 83iT^{2} \)
89 \( 1 + 7.60T + 89T^{2} \)
97 \( 1 - 9.70iT - 97T^{2} \)
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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

−10.13662660840706566065612033522, −9.851801032914220606119850767889, −8.249635823781855220163900740813, −7.81345025422708344392753761129, −7.04623944247309278739546230656, −6.11969296043171706922290368503, −4.30211884245971873982528552823, −3.52185103348482568200566552735, −2.39567625784763021010968261166, −0.65028854698001946277749094153, 1.20434219137634980474840093488, 2.67266187931989257809748733987, 4.31807762025996382300602586797, 5.59320181268568078349910052741, 5.96604660869276383240552858038, 7.39381371713942163768412102391, 8.173028279883942363499029193201, 8.762382521457129222816227635302, 9.449135009438299808821131343119, 10.40051919185343179473195533556

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