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Properties

Label 2-360-120.59-c3-0-29
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 + 29.1·7-s + 22.6i·8-s − 31.6·10-s + 72.3i·11-s + 32.0·13-s − 82.4i·14-s + 64.0·16-s + 107.·19-s + 89.4i·20-s + 204.·22-s + 219. i·23-s − 125.·25-s − 90.7i·26-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s − 0.999i·5-s + 1.57·7-s + 1.00i·8-s − 1.00·10-s + 1.98i·11-s + 0.684·13-s − 1.57i·14-s + 1.00·16-s + 1.29·19-s + 1.00i·20-s + 1.98·22-s + 1.98i·23-s − 1.00·25-s − 0.684i·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

\(L(2)\) \(\approx\) \(2.086996387\)
\(L(\frac12)\) \(\approx\) \(2.086996387\)
\(L(\frac{5}{2})\) 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 \( 1 - 29.1T + 343T^{2} \)
11 \( 1 - 72.3iT - 1.33e3T^{2} \)
13 \( 1 - 32.0T + 2.19e3T^{2} \)
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 \( 1 - 151.T + 5.06e4T^{2} \)
41 \( 1 + 162. iT - 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 - 376. iT - 1.03e5T^{2} \)
53 \( 1 + 718. iT - 1.48e5T^{2} \)
59 \( 1 + 800. iT - 2.05e5T^{2} \)
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 \( 1 - 1.65e3iT - 7.04e5T^{2} \)
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

−11.15911211424959981154761339310, −9.811942600575782760469281290788, −9.327672776133034870711775404924, −8.140007179696916476215538016576, −7.53354897288894212379139140735, −5.35879944591083226437754635501, −4.85238120859744064582077775505, −3.85341643231277167009508827274, −1.91686217325279382816826287545, −1.26127892594647815107714884204, 0.922652673779614722989804395550, 3.03641078058904323912947709169, 4.29441804830868340862057252981, 5.56188267215791193190600384483, 6.26692893140143239626026253804, 7.45306308353804672953245206408, 8.257417278448363460732666494705, 8.860242162475031966360625491313, 10.37358191971498292301622897806, 11.06512098178667585150561546535

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