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Properties

Label 2-360-120.53-c1-0-14
Degree $2$
Conductor $360$
Sign $0.0269 + 0.999i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
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  + (−0.803 − 1.16i)2-s + (−0.709 + 1.87i)4-s + (1.29 − 1.82i)5-s + (−0.306 + 0.306i)7-s + (2.74 − 0.676i)8-s + (−3.16 − 0.0386i)10-s + 4.06·11-s + (−0.625 + 0.625i)13-s + (0.603 + 0.110i)14-s + (−2.99 − 2.65i)16-s + (−3.57 − 3.57i)17-s + 6.82·19-s + (2.49 + 3.71i)20-s + (−3.26 − 4.73i)22-s + (1.58 − 1.58i)23-s + ⋯
L(s)  = 1  + (−0.568 − 0.822i)2-s + (−0.354 + 0.935i)4-s + (0.578 − 0.815i)5-s + (−0.115 + 0.115i)7-s + (0.970 − 0.239i)8-s + (−0.999 − 0.0122i)10-s + 1.22·11-s + (−0.173 + 0.173i)13-s + (0.161 + 0.0295i)14-s + (−0.748 − 0.663i)16-s + (−0.867 − 0.867i)17-s + 1.56·19-s + (0.557 + 0.829i)20-s + (−0.696 − 1.00i)22-s + (0.331 − 0.331i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0269 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0269 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.0269 + 0.999i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.0269 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.792450 - 0.771350i\)
\(L(\frac12)\) \(\approx\) \(0.792450 - 0.771350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.803 + 1.16i)T \)
3 \( 1 \)
5 \( 1 + (-1.29 + 1.82i)T \)
good7 \( 1 + (0.306 - 0.306i)T - 7iT^{2} \)
11 \( 1 - 4.06T + 11T^{2} \)
13 \( 1 + (0.625 - 0.625i)T - 13iT^{2} \)
17 \( 1 + (3.57 + 3.57i)T + 17iT^{2} \)
19 \( 1 - 6.82T + 19T^{2} \)
23 \( 1 + (-1.58 + 1.58i)T - 23iT^{2} \)
29 \( 1 + 8.50iT - 29T^{2} \)
31 \( 1 + 2.56T + 31T^{2} \)
37 \( 1 + (-1.69 - 1.69i)T + 37iT^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 + (-4.18 + 4.18i)T - 43iT^{2} \)
47 \( 1 + (4.58 + 4.58i)T + 47iT^{2} \)
53 \( 1 + (7.41 + 7.41i)T + 53iT^{2} \)
59 \( 1 - 8.79iT - 59T^{2} \)
61 \( 1 - 6.08iT - 61T^{2} \)
67 \( 1 + (-6.18 - 6.18i)T + 67iT^{2} \)
71 \( 1 - 14.7iT - 71T^{2} \)
73 \( 1 + (-3.05 - 3.05i)T + 73iT^{2} \)
79 \( 1 + 8.56iT - 79T^{2} \)
83 \( 1 + (-5.13 - 5.13i)T + 83iT^{2} \)
89 \( 1 - 9.88T + 89T^{2} \)
97 \( 1 + (-1.75 + 1.75i)T - 97iT^{2} \)
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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

−11.52160222109982068818624041575, −10.03986956761035053747603693498, −9.375067332893445160202678510412, −8.868415923173471514534597601136, −7.67170838671225827800788637126, −6.50905684615301568268240898727, −5.06181597614027245473512205779, −4.01151162295440501284365723643, −2.49626569411161701778996018241, −1.06657046505835766927151273404, 1.61724679174773874940213426669, 3.51088175117079812118565873269, 5.07406498350182463244649603241, 6.19307032104188566078061157700, 6.85914712444448941106613416301, 7.74333967573236964936659356084, 9.114554186349791252610815825910, 9.526640496804205126523153971523, 10.65747474252218397910132282691, 11.27423043386696634991816245450

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