Properties

Label 2-360-15.8-c1-0-2
Degree $2$
Conductor $360$
Sign $0.920 - 0.391i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.920 - 0.391i$
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: \((2,\ 360,\ (\ :1/2),\ 0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54930 + 0.315465i\)
\(L(\frac12)\) \(\approx\) \(1.54930 + 0.315465i\)
\(L(\frac{3}{2})\) 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

−11.41988703971922783276325949264, −10.39784893023153992957445056146, −9.961425809699818800499867931479, −8.732444003793868112108894414845, −7.62277009109294712992540671011, −6.75909006982504014277243160983, −5.72997979725449398990759510679, −4.53789091013183312405727055079, −3.15111566919436405490778005006, −1.70003351839118246004774963364, 1.37278999509265158877939824350, 2.92657229829980087705000901642, 4.56601842448959397473806464142, 5.53800099660868273684026101433, 6.32508402981278647292462063106, 7.939162412157422278339488161054, 8.476230518548458611585679671905, 9.669055315559551452099850129732, 10.21687800668154261828375339007, 11.62477299331103663466312423785

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