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Properties

Label 2-360-40.3-c1-0-27
Degree $2$
Conductor $360$
Sign $-0.999 + 0.0290i$
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  + (0.804 − 1.16i)2-s + (−0.705 − 1.87i)4-s + (−1.51 + 1.64i)5-s + (−3.43 − 3.43i)7-s + (−2.74 − 0.684i)8-s + (0.696 + 3.08i)10-s − 3.48·11-s + (2.05 − 2.05i)13-s + (−6.76 + 1.23i)14-s + (−3.00 + 2.64i)16-s + (1.64 − 1.64i)17-s − 0.642i·19-s + (4.14 + 1.67i)20-s + (−2.80 + 4.04i)22-s + (2.31 − 2.31i)23-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)2-s + (−0.352 − 0.935i)4-s + (−0.676 + 0.736i)5-s + (−1.29 − 1.29i)7-s + (−0.970 − 0.242i)8-s + (0.220 + 0.975i)10-s − 1.04·11-s + (0.568 − 0.568i)13-s + (−1.80 + 0.329i)14-s + (−0.751 + 0.660i)16-s + (0.400 − 0.400i)17-s − 0.147i·19-s + (0.927 + 0.373i)20-s + (−0.597 + 0.863i)22-s + (0.481 − 0.481i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.999 + 0.0290i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.999 + 0.0290i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0120707 - 0.831891i\)
\(L(\frac12)\) \(\approx\) \(0.0120707 - 0.831891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.804 + 1.16i)T \)
3 \( 1 \)
5 \( 1 + (1.51 - 1.64i)T \)
good7 \( 1 + (3.43 + 3.43i)T + 7iT^{2} \)
11 \( 1 + 3.48T + 11T^{2} \)
13 \( 1 + (-2.05 + 2.05i)T - 13iT^{2} \)
17 \( 1 + (-1.64 + 1.64i)T - 17iT^{2} \)
19 \( 1 + 0.642iT - 19T^{2} \)
23 \( 1 + (-2.31 + 2.31i)T - 23iT^{2} \)
29 \( 1 - 0.699T + 29T^{2} \)
31 \( 1 + 1.56iT - 31T^{2} \)
37 \( 1 + (-5.31 - 5.31i)T + 37iT^{2} \)
41 \( 1 + 4.92T + 41T^{2} \)
43 \( 1 + (-3.56 - 3.56i)T + 43iT^{2} \)
47 \( 1 + (6.85 + 6.85i)T + 47iT^{2} \)
53 \( 1 + (-1.94 + 1.94i)T - 53iT^{2} \)
59 \( 1 - 2.74iT - 59T^{2} \)
61 \( 1 + 5.20iT - 61T^{2} \)
67 \( 1 + (-6.92 + 6.92i)T - 67iT^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + (6.56 + 6.56i)T + 73iT^{2} \)
79 \( 1 + 2.09T + 79T^{2} \)
83 \( 1 + (6.64 + 6.64i)T + 83iT^{2} \)
89 \( 1 - 0.733iT - 89T^{2} \)
97 \( 1 + (-8.79 + 8.79i)T - 97iT^{2} \)
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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

−10.82585783006027409723027500149, −10.38495546559427159939606961104, −9.631075667356574166418472909767, −8.092330853818986232250020076335, −7.01407214365007157095464611882, −6.14514769574881166933818605859, −4.69224590912424689536170505291, −3.50949585396748993528417187301, −2.92144502723229417443962247764, −0.45909988899390546058436337504, 2.84282594065705050107542135613, 3.92945140382677943852458963421, 5.27239755297855600605634262333, 5.93732370379407696292550695361, 7.07563041441728974778430006700, 8.197487763300818533250831941267, 8.863692573102748880432312889641, 9.713645506076138834893779992653, 11.33111015671421298332857889358, 12.24487887513239868223034768618

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