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Properties

Label 2-360-120.77-c1-0-13
Degree $2$
Conductor $360$
Sign $0.987 + 0.157i$
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  + (0.159 + 1.40i)2-s + (−1.94 + 0.449i)4-s + (0.215 − 2.22i)5-s + (−2.32 − 2.32i)7-s + (−0.943 − 2.66i)8-s + (3.16 − 0.0533i)10-s + 5.57·11-s + (1.79 + 1.79i)13-s + (2.88 − 3.63i)14-s + (3.59 − 1.75i)16-s + (5.56 − 5.56i)17-s − 2.61·19-s + (0.580 + 4.43i)20-s + (0.892 + 7.83i)22-s + (−4.44 − 4.44i)23-s + ⋯
L(s)  = 1  + (0.113 + 0.993i)2-s + (−0.974 + 0.224i)4-s + (0.0963 − 0.995i)5-s + (−0.877 − 0.877i)7-s + (−0.333 − 0.942i)8-s + (0.999 − 0.0168i)10-s + 1.68·11-s + (0.497 + 0.497i)13-s + (0.772 − 0.970i)14-s + (0.898 − 0.438i)16-s + (1.34 − 1.34i)17-s − 0.600·19-s + (0.129 + 0.991i)20-s + (0.190 + 1.67i)22-s + (−0.927 − 0.927i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.987 + 0.157i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.987 + 0.157i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20829 - 0.0957512i\)
\(L(\frac12)\) \(\approx\) \(1.20829 - 0.0957512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.159 - 1.40i)T \)
3 \( 1 \)
5 \( 1 + (-0.215 + 2.22i)T \)
good7 \( 1 + (2.32 + 2.32i)T + 7iT^{2} \)
11 \( 1 - 5.57T + 11T^{2} \)
13 \( 1 + (-1.79 - 1.79i)T + 13iT^{2} \)
17 \( 1 + (-5.56 + 5.56i)T - 17iT^{2} \)
19 \( 1 + 2.61T + 19T^{2} \)
23 \( 1 + (4.44 + 4.44i)T + 23iT^{2} \)
29 \( 1 - 2.12iT - 29T^{2} \)
31 \( 1 + 4.52T + 31T^{2} \)
37 \( 1 + (-5.02 + 5.02i)T - 37iT^{2} \)
41 \( 1 - 1.73iT - 41T^{2} \)
43 \( 1 + (-1.19 - 1.19i)T + 43iT^{2} \)
47 \( 1 + (0.849 - 0.849i)T - 47iT^{2} \)
53 \( 1 + (-4.22 + 4.22i)T - 53iT^{2} \)
59 \( 1 - 8.08iT - 59T^{2} \)
61 \( 1 - 3.13iT - 61T^{2} \)
67 \( 1 + (1.86 - 1.86i)T - 67iT^{2} \)
71 \( 1 - 6.95iT - 71T^{2} \)
73 \( 1 + (5.86 - 5.86i)T - 73iT^{2} \)
79 \( 1 - 2.52iT - 79T^{2} \)
83 \( 1 + (-0.694 + 0.694i)T - 83iT^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-4.09 - 4.09i)T + 97iT^{2} \)
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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

−11.71127750091697695806562499515, −10.07038923956135688172431254187, −9.362277993314851833695832701075, −8.699221902864179339068499940492, −7.48580536158072536173234608038, −6.62040524649196751842569673005, −5.75665170975333443444381172792, −4.38596772551189097693410426325, −3.71956029505605802608231839704, −0.895228784758486374713210786868, 1.79186998627450447018997698499, 3.27907706223147849036435733835, 3.85938176430693987960931931493, 5.86039751129206422387324826027, 6.24780260291475933095614514274, 7.910349025407858276562442157269, 9.038317568439330056113322641042, 9.790822297368763497385998515385, 10.53009043158074127855885395313, 11.57032426703175696825527487035

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