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Properties

Label 2-360-40.27-c1-0-13
Degree $2$
Conductor $360$
Sign $0.891 - 0.453i$
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  + (0.221 + 1.39i)2-s + (−1.90 + 0.618i)4-s + (1.17 − 1.90i)5-s + (1.90 − 1.90i)7-s + (−1.28 − 2.52i)8-s + (2.91 + 1.22i)10-s + 3.23·11-s + (−0.726 − 0.726i)13-s + (3.07 + 2.23i)14-s + (3.23 − 2.35i)16-s + (1 + i)17-s − 2i·19-s + (−1.06 + 4.34i)20-s + (0.715 + 4.52i)22-s + (4.25 + 4.25i)23-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)2-s + (−0.951 + 0.309i)4-s + (0.525 − 0.850i)5-s + (0.718 − 0.718i)7-s + (−0.453 − 0.891i)8-s + (0.922 + 0.386i)10-s + 0.975·11-s + (−0.201 − 0.201i)13-s + (0.822 + 0.597i)14-s + (0.809 − 0.587i)16-s + (0.242 + 0.242i)17-s − 0.458i·19-s + (−0.237 + 0.971i)20-s + (0.152 + 0.963i)22-s + (0.886 + 0.886i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.891 - 0.453i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.891 - 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50950 + 0.362399i\)
\(L(\frac12)\) \(\approx\) \(1.50950 + 0.362399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.221 - 1.39i)T \)
3 \( 1 \)
5 \( 1 + (-1.17 + 1.90i)T \)
good7 \( 1 + (-1.90 + 1.90i)T - 7iT^{2} \)
11 \( 1 - 3.23T + 11T^{2} \)
13 \( 1 + (0.726 + 0.726i)T + 13iT^{2} \)
17 \( 1 + (-1 - i)T + 17iT^{2} \)
19 \( 1 + 2iT - 19T^{2} \)
23 \( 1 + (-4.25 - 4.25i)T + 23iT^{2} \)
29 \( 1 + 6.15T + 29T^{2} \)
31 \( 1 - 8.50iT - 31T^{2} \)
37 \( 1 + (-0.726 + 0.726i)T - 37iT^{2} \)
41 \( 1 + 5.70T + 41T^{2} \)
43 \( 1 + (-4.61 + 4.61i)T - 43iT^{2} \)
47 \( 1 + (-3.35 + 3.35i)T - 47iT^{2} \)
53 \( 1 + (3.07 + 3.07i)T + 53iT^{2} \)
59 \( 1 - 0.472iT - 59T^{2} \)
61 \( 1 + 0.898iT - 61T^{2} \)
67 \( 1 + (4.61 + 4.61i)T + 67iT^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (4.70 - 4.70i)T - 73iT^{2} \)
79 \( 1 + 2.90T + 79T^{2} \)
83 \( 1 + (-6.61 + 6.61i)T - 83iT^{2} \)
89 \( 1 + 2.47iT - 89T^{2} \)
97 \( 1 + (-4.23 - 4.23i)T + 97iT^{2} \)
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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

−11.66861078478206941731444922161, −10.41063531567298730401458357649, −9.314705649198891177649218340985, −8.730845019442835716695226901161, −7.64789109178907228429352056954, −6.80390176474097542758067662395, −5.56695520162272704610009710773, −4.79858600404903128705383254023, −3.72688406204262459653840886439, −1.26476519738723477677176822607, 1.72790732431045930482965840515, 2.82370774028713957140414369073, 4.14539672535001973180135470108, 5.39038999632128707983450944441, 6.34033119954016537375803470742, 7.74321178630606147372369434094, 8.993373406076395543879676608144, 9.576156577589139111376373050695, 10.64743120959013420927416213995, 11.39744288001080489829827430573

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