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Properties

Label 2-360-120.59-c3-0-3
Degree $2$
Conductor $360$
Sign $-0.577 - 0.816i$
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  − 2.82i·2-s − 8.00·4-s + 11.1i·5-s + 22.8·7-s + 22.6i·8-s + 31.6·10-s + 9.70i·11-s − 88.0·13-s − 64.5i·14-s + 64.0·16-s − 107.·19-s − 89.4i·20-s + 27.4·22-s − 219. i·23-s − 125.·25-s + 249. i·26-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + 0.999i·5-s + 1.23·7-s + 1.00i·8-s + 1.00·10-s + 0.266i·11-s − 1.87·13-s − 1.23i·14-s + 1.00·16-s − 1.29·19-s − 1.00i·20-s + 0.266·22-s − 1.98i·23-s − 1.00·25-s + 1.87i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.577 - 0.816i$
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: \((2,\ 360,\ (\ :3/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2168397103\)
\(L(\frac12)\) \(\approx\) \(0.2168397103\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2.82iT \)
3 \( 1 \)
5 \( 1 - 11.1iT \)
good7 \( 1 - 22.8T + 343T^{2} \)
11 \( 1 - 9.70iT - 1.33e3T^{2} \)
13 \( 1 + 88.0T + 2.19e3T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 + 107.T + 6.85e3T^{2} \)
23 \( 1 + 219. iT - 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 + 423.T + 5.06e4T^{2} \)
41 \( 1 - 499. iT - 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 - 376. iT - 1.03e5T^{2} \)
53 \( 1 + 718. iT - 1.48e5T^{2} \)
59 \( 1 - 424. iT - 2.05e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 - 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 254. iT - 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−11.24942752123629629366848576765, −10.49288549261469220604794890986, −9.892570067087680631625602580843, −8.597528869237921045212932215329, −7.76852390723261625325921702361, −6.62128975976114627913276962002, −5.02606984530348083337322162475, −4.33339846517570715697679009623, −2.73316417439033180441380598737, −1.95197434245701923340090596387, 0.07193540005558403078397262133, 1.76989176876362452545031142554, 4.01350593255900351479783822460, 5.03597677555588387477679795325, 5.49062553545156846192012660813, 7.09760604562985065016207446965, 7.83900630363273741420784733255, 8.670961883471221763164670451178, 9.438756703313815920179394215601, 10.52007011826603757349251844279

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