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Properties

Label 2-360-120.59-c3-0-16
Degree $2$
Conductor $360$
Sign $-0.368 - 0.929i$
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  + (−2.59 − 1.12i)2-s + (5.45 + 5.85i)4-s + (6.79 + 8.88i)5-s − 14.0·7-s + (−7.54 − 21.3i)8-s + (−7.59 − 30.6i)10-s + 30.3i·11-s + 63.3·13-s + (36.4 + 15.8i)14-s + (−4.50 + 63.8i)16-s − 111.·17-s + 151.·19-s + (−14.9 + 88.1i)20-s + (34.2 − 78.8i)22-s + 48.6i·23-s + ⋯
L(s)  = 1  + (−0.916 − 0.398i)2-s + (0.681 + 0.731i)4-s + (0.607 + 0.794i)5-s − 0.758·7-s + (−0.333 − 0.942i)8-s + (−0.240 − 0.970i)10-s + 0.833i·11-s + 1.35·13-s + (0.695 + 0.302i)14-s + (−0.0704 + 0.997i)16-s − 1.59·17-s + 1.82·19-s + (−0.166 + 0.985i)20-s + (0.332 − 0.763i)22-s + 0.440i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.368 - 0.929i$
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: \((2,\ 360,\ (\ :3/2),\ -0.368 - 0.929i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8483152834\)
\(L(\frac12)\) \(\approx\) \(0.8483152834\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.59 + 1.12i)T \)
3 \( 1 \)
5 \( 1 + (-6.79 - 8.88i)T \)
good7 \( 1 + 14.0T + 343T^{2} \)
11 \( 1 - 30.3iT - 1.33e3T^{2} \)
13 \( 1 - 63.3T + 2.19e3T^{2} \)
17 \( 1 + 111.T + 4.91e3T^{2} \)
19 \( 1 - 151.T + 6.85e3T^{2} \)
23 \( 1 - 48.6iT - 1.21e4T^{2} \)
29 \( 1 + 140.T + 2.43e4T^{2} \)
31 \( 1 + 148. iT - 2.97e4T^{2} \)
37 \( 1 + 313.T + 5.06e4T^{2} \)
41 \( 1 - 317. iT - 6.89e4T^{2} \)
43 \( 1 + 185. iT - 7.95e4T^{2} \)
47 \( 1 + 5.81iT - 1.03e5T^{2} \)
53 \( 1 - 11.0iT - 1.48e5T^{2} \)
59 \( 1 - 487. iT - 2.05e5T^{2} \)
61 \( 1 - 615. iT - 2.26e5T^{2} \)
67 \( 1 - 1.05e3iT - 3.00e5T^{2} \)
71 \( 1 + 731.T + 3.57e5T^{2} \)
73 \( 1 - 712. iT - 3.89e5T^{2} \)
79 \( 1 + 1.18e3iT - 4.93e5T^{2} \)
83 \( 1 - 571.T + 5.71e5T^{2} \)
89 \( 1 - 210. iT - 7.04e5T^{2} \)
97 \( 1 - 796. iT - 9.12e5T^{2} \)
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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

−11.16006257156330743818060020762, −10.25682943489407221655604776830, −9.538523633771562349375117604162, −8.838169569046682468408772915779, −7.42913834330153301584694847203, −6.77894541356359361500816706715, −5.80316151473618745091030539066, −3.82957541029744045877751982382, −2.78771739732616078048643578587, −1.53480676053389257001509841508, 0.40038181362912678868391182194, 1.67943316230258633919611090931, 3.33122665202585669467741386024, 5.14551517270266845541677589691, 6.06616023876421230641054252330, 6.81914198730651663341124762861, 8.197386237222360120873250691448, 8.974646455868842655997988061757, 9.470316009176330921883491064827, 10.62533836575902193440187967395

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