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Properties

Label 2-360-120.59-c3-0-21
Degree $2$
Conductor $360$
Sign $-0.519 - 0.854i$
Analytic cond. $21.2406$
Root an. cond. $4.60876$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.33 + 1.59i)2-s + (2.89 − 7.45i)4-s + (−3.30 + 10.6i)5-s + 10.1·7-s + (5.17 + 22.0i)8-s + (−9.35 − 30.2i)10-s + 13.8i·11-s + 41.4·13-s + (−23.7 + 16.2i)14-s + (−47.2 − 43.1i)16-s + 27.7·17-s + 15.2·19-s + (70.1 + 55.5i)20-s + (−22.1 − 32.3i)22-s − 3.27i·23-s + ⋯
L(s)  = 1  + (−0.825 + 0.565i)2-s + (0.361 − 0.932i)4-s + (−0.295 + 0.955i)5-s + 0.548·7-s + (0.228 + 0.973i)8-s + (−0.295 − 0.955i)10-s + 0.380i·11-s + 0.884·13-s + (−0.452 + 0.309i)14-s + (−0.738 − 0.674i)16-s + 0.395·17-s + 0.184·19-s + (0.783 + 0.621i)20-s + (−0.214 − 0.313i)22-s − 0.0296i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.519 - 0.854i$
Analytic conductor: \(21.2406\)
Root analytic conductor: \(4.60876\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :3/2),\ -0.519 - 0.854i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.140815638\)
\(L(\frac12)\) \(\approx\) \(1.140815638\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.33 - 1.59i)T \)
3 \( 1 \)
5 \( 1 + (3.30 - 10.6i)T \)
good7 \( 1 - 10.1T + 343T^{2} \)
11 \( 1 - 13.8iT - 1.33e3T^{2} \)
13 \( 1 - 41.4T + 2.19e3T^{2} \)
17 \( 1 - 27.7T + 4.91e3T^{2} \)
19 \( 1 - 15.2T + 6.85e3T^{2} \)
23 \( 1 + 3.27iT - 1.21e4T^{2} \)
29 \( 1 - 100.T + 2.43e4T^{2} \)
31 \( 1 - 8.55iT - 2.97e4T^{2} \)
37 \( 1 - 190.T + 5.06e4T^{2} \)
41 \( 1 - 211. iT - 6.89e4T^{2} \)
43 \( 1 - 398. iT - 7.95e4T^{2} \)
47 \( 1 - 171. iT - 1.03e5T^{2} \)
53 \( 1 - 124. iT - 1.48e5T^{2} \)
59 \( 1 - 354. iT - 2.05e5T^{2} \)
61 \( 1 + 397. iT - 2.26e5T^{2} \)
67 \( 1 - 554. iT - 3.00e5T^{2} \)
71 \( 1 + 961.T + 3.57e5T^{2} \)
73 \( 1 - 920. iT - 3.89e5T^{2} \)
79 \( 1 + 683. iT - 4.93e5T^{2} \)
83 \( 1 + 1.15e3T + 5.71e5T^{2} \)
89 \( 1 - 581. iT - 7.04e5T^{2} \)
97 \( 1 - 1.34e3iT - 9.12e5T^{2} \)
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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

−11.15202213102375539396140622421, −10.35081208440937858701871636854, −9.501751446723379802323277473229, −8.307714210649548334810123078104, −7.67683855880822720501351177518, −6.68751417626289489385596439027, −5.84565725651921562390049074482, −4.45918013285843655027977906973, −2.83046714512082591230658737809, −1.30377570292555654194227635586, 0.59385390032264420934977884247, 1.70810693861698652294906143713, 3.37816955882731908839730365152, 4.48134669285585920749270539642, 5.82157684988962213094358122294, 7.25973613569604542224295947332, 8.267815929728933055361782773243, 8.712683102699375707680330887842, 9.726694492269320056589949313702, 10.76132880190801962751812141879

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