Properties

Label 2-360-5.4-c1-0-4
Degree $2$
Conductor $360$
Sign $0.894 + 0.447i$
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 − 2i)5-s + 2i·7-s + 4·11-s − 4i·13-s + 4·19-s − 2i·23-s + (−3 − 4i)25-s + 2·29-s + (4 + 2i)35-s + 4i·37-s − 2·41-s + 6i·43-s + 6i·47-s + 3·49-s − 4i·53-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)5-s + 0.755i·7-s + 1.20·11-s − 1.10i·13-s + 0.917·19-s − 0.417i·23-s + (−0.600 − 0.800i)25-s + 0.371·29-s + (0.676 + 0.338i)35-s + 0.657i·37-s − 0.312·41-s + 0.914i·43-s + 0.875i·47-s + 0.428·49-s − 0.549i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47492 - 0.348182i\)
\(L(\frac12)\) \(\approx\) \(1.47492 - 0.348182i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 14iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−11.62146706578789122244066694284, −10.30702392713714639753250898354, −9.396936007551214622678105567950, −8.743542844396008738709760611756, −7.78163306240406971436674874828, −6.35013980618183755690568839468, −5.54004595599785564699608818615, −4.50006638623930153797112900260, −2.97156067584272290999047438192, −1.30437555056029370875722557072, 1.66107779292882608091093468010, 3.32051266049004067327964353717, 4.34534625404125172218814009652, 5.87617307431092642396527985062, 6.84725379222898699696651587228, 7.42096671005332491124093741311, 8.962362085644337362446530658841, 9.683191293094794001797824238203, 10.60889806535686417703199682791, 11.45985362797018267468215694097

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