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Properties

Label 2-360-120.59-c1-0-15
Degree $2$
Conductor $360$
Sign $0.741 - 0.671i$
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  + (1.37 + 0.331i)2-s + (1.78 + 0.910i)4-s + (1.64 + 1.51i)5-s − 0.936·7-s + (2.14 + 1.84i)8-s + (1.76 + 2.62i)10-s − 2.20i·11-s − 3.33·13-s + (−1.28 − 0.310i)14-s + (2.34 + 3.24i)16-s + 1.54·17-s − 3.12·19-s + (1.56 + 4.19i)20-s + (0.731 − 3.03i)22-s − 3.39i·23-s + ⋯
L(s)  = 1  + (0.972 + 0.234i)2-s + (0.890 + 0.455i)4-s + (0.737 + 0.675i)5-s − 0.353·7-s + (0.759 + 0.650i)8-s + (0.558 + 0.829i)10-s − 0.665i·11-s − 0.924·13-s + (−0.344 − 0.0828i)14-s + (0.585 + 0.810i)16-s + 0.374·17-s − 0.716·19-s + (0.349 + 0.937i)20-s + (0.155 − 0.647i)22-s − 0.707i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.741 - 0.671i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.741 - 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.34975 + 0.905818i\)
\(L(\frac12)\) \(\approx\) \(2.34975 + 0.905818i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.331i)T \)
3 \( 1 \)
5 \( 1 + (-1.64 - 1.51i)T \)
good7 \( 1 + 0.936T + 7T^{2} \)
11 \( 1 + 2.20iT - 11T^{2} \)
13 \( 1 + 3.33T + 13T^{2} \)
17 \( 1 - 1.54T + 17T^{2} \)
19 \( 1 + 3.12T + 19T^{2} \)
23 \( 1 + 3.39iT - 23T^{2} \)
29 \( 1 - 8.44T + 29T^{2} \)
31 \( 1 + 8.30iT - 31T^{2} \)
37 \( 1 + 7.60T + 37T^{2} \)
41 \( 1 + 5.83iT - 41T^{2} \)
43 \( 1 - 7.77iT - 43T^{2} \)
47 \( 1 + 10.7iT - 47T^{2} \)
53 \( 1 - 5.08iT - 53T^{2} \)
59 \( 1 - 10.6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 12.1iT - 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 + 5.59iT - 73T^{2} \)
79 \( 1 + 1.02iT - 79T^{2} \)
83 \( 1 + 14.0T + 83T^{2} \)
89 \( 1 - 13.0iT - 89T^{2} \)
97 \( 1 + 2.18iT - 97T^{2} \)
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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

−11.71835223039918664832713241290, −10.64773394093903142654418514403, −10.02524955210246665656122988625, −8.656742853603209863127287921754, −7.45303382238231881949830230136, −6.52079188617424222598819391206, −5.82987009101288303434848899451, −4.65912171807924272651512771495, −3.26916417347108720853729656237, −2.28957179745882660202051676404, 1.67976700410473206441350790468, 2.98926117330532398673005101753, 4.52578541123784101277851868561, 5.22616522637484765938070806832, 6.34543639495012365546543748269, 7.23130498159394045672618810997, 8.622429038391097595415286445580, 9.898430377481666419966523329294, 10.24242431875739008994523394472, 11.62974393478312127335396716085

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