Properties

Label 2-360-15.8-c1-0-4
Degree $2$
Conductor $360$
Sign $0.0618 + 0.998i$
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  + (−1.58 − 1.58i)5-s + (1.23 − 1.23i)7-s − 1.74i·11-s + (0.236 + 0.236i)13-s + (−4.57 − 4.57i)17-s − 6.47i·19-s + (2.82 − 2.82i)23-s + 5.00i·25-s − 0.333·29-s + 10.4·31-s − 3.90·35-s + (−2.23 + 2.23i)37-s + 7.07i·41-s + (−6.47 − 6.47i)43-s + (4.57 + 4.57i)47-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)5-s + (0.467 − 0.467i)7-s − 0.527i·11-s + (0.0654 + 0.0654i)13-s + (−1.10 − 1.10i)17-s − 1.48i·19-s + (0.589 − 0.589i)23-s + 1.00i·25-s − 0.0619·29-s + 1.88·31-s − 0.660·35-s + (−0.367 + 0.367i)37-s + 1.10i·41-s + (−0.986 − 0.986i)43-s + (0.667 + 0.667i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.0618 + 0.998i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.0618 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.799305 - 0.751294i\)
\(L(\frac12)\) \(\approx\) \(0.799305 - 0.751294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (1.58 + 1.58i)T \)
good7 \( 1 + (-1.23 + 1.23i)T - 7iT^{2} \)
11 \( 1 + 1.74iT - 11T^{2} \)
13 \( 1 + (-0.236 - 0.236i)T + 13iT^{2} \)
17 \( 1 + (4.57 + 4.57i)T + 17iT^{2} \)
19 \( 1 + 6.47iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 + 0.333T + 29T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + (2.23 - 2.23i)T - 37iT^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + (6.47 + 6.47i)T + 43iT^{2} \)
47 \( 1 + (-4.57 - 4.57i)T + 47iT^{2} \)
53 \( 1 - 53iT^{2} \)
59 \( 1 - 7.40T + 59T^{2} \)
61 \( 1 - 1.52T + 61T^{2} \)
67 \( 1 + (10.4 - 10.4i)T - 67iT^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + (9.47 + 9.47i)T + 73iT^{2} \)
79 \( 1 + 5.52iT - 79T^{2} \)
83 \( 1 + (-7.40 + 7.40i)T - 83iT^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (-1 + i)T - 97iT^{2} \)
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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

−11.43002101523839753001402393471, −10.45496303231410471801413933437, −9.101198762105403521362655649553, −8.552751664070042658991986557148, −7.47147741864327302035861138309, −6.57435809446095242257149880712, −4.95006272674043062342521509716, −4.41553447727808354651167594360, −2.84444473672419754934912943438, −0.77958564745287889848075972126, 2.02681507606088589201715121493, 3.52186598477977075525965873129, 4.58951536271278105188430941958, 5.95824870213647930440679055413, 6.93909698607323915837064543468, 7.989996833741295672997203050184, 8.667527932473993877009520422013, 10.03485628586878977511344754376, 10.75208716675501741814334437397, 11.70707372943953873319503643365

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