Properties

Label 2-360-120.59-c1-0-9
Degree $2$
Conductor $360$
Sign $0.577 - 0.816i$
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  + 1.41i·2-s − 2.00·4-s − 2.23i·5-s + 1.16·7-s − 2.82i·8-s + 3.16·10-s + 5.88i·11-s + 7.16·13-s + 1.64i·14-s + 4.00·16-s + 6.32·19-s + 4.47i·20-s − 8.32·22-s − 4.47i·23-s − 5.00·25-s + 10.1i·26-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s − 0.999i·5-s + 0.439·7-s − 1.00i·8-s + 1.00·10-s + 1.77i·11-s + 1.98·13-s + 0.439i·14-s + 1.00·16-s + 1.45·19-s + 1.00i·20-s − 1.77·22-s − 0.932i·23-s − 1.00·25-s + 1.98i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.577 - 0.816i$
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: \((2,\ 360,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20129 + 0.621838i\)
\(L(\frac12)\) \(\approx\) \(1.20129 + 0.621838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
5 \( 1 + 2.23iT \)
good7 \( 1 - 1.16T + 7T^{2} \)
11 \( 1 - 5.88iT - 11T^{2} \)
13 \( 1 - 7.16T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 6.32T + 19T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 4.83T + 37T^{2} \)
41 \( 1 + 7.53iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 2.82iT - 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 - 14.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 0.955iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.82371335409529944941038531603, −10.42701899616266383649391610364, −9.424047137569716231696992770556, −8.682357014128935249360882932533, −7.85652118160776739448918885789, −6.88331381211836733152531048145, −5.68511512222194598575690538820, −4.80588587888937454904143072103, −3.89100685333559746365301111849, −1.36474661679798379689652479566, 1.31346534934583332374835228493, 3.17269742691337179327917510861, 3.63710201170448765273606466771, 5.41382295049435944719099842825, 6.26264692259677652491296357441, 7.85783261441437572162572179951, 8.598286009848347184001098499254, 9.629709061325891906310094833756, 10.77572798978005727751908412946, 11.22834739878756709437209641795

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